diff --git "a/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/load_file.txt" "b/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-dAzT4oBgHgl3EQfg_zf/content/tmp_files/load_file.txt" @@ -0,0 +1,874 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf,len=873 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='01479v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='OC] 4 Jan 2023 Generalizations of R0 and SSM properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Extended Horizontal Linear Complementarity Problem Punit Kumar Yadav Department of Mathematics Malaviya National Instiute of Technology, Jaipur, 302017, India E-mail address: punitjrf@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='com K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Palpandi Department of Mathematics Malaviya National Instiute of Technology, Jaipur, 302017, India E-mail address: kpalpandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='maths@mnit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='in Abstract In this paper, we first introduce R0-W and SSM-W property for the set of matrices which is a generalization of R0 and the strictly semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We then prove some existence results for the extended horizontal linear complementarity problem when the involved matrices have these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' With an additional condition on the set of matrices, we prove that the SSM-W property is equivalent to the unique solution for the corresponding extended horizontal linear complementarity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Finally, we give a necessary and sufficient condition for the connectedness of the solution set of the extended horizontal linear complementarity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 1 Introduction The standard linear complementarity problem (for short LCP), LCP(C, q), is to find vectors x, y such that x ∈ Rn, y = Cx + q ∈ Rn and x ∧ y = 0, (1) where C ∈ Rn×n, q ∈ Rn and ′∧′ is a min map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The LCP has numerous applications in numerous domains, such as optimization, economics, and game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Cottle and Pang’s monograph [1] is the primary reference for standard LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Various generalisations of the linear complementarity problem have been developed and discussed in the literature during the past three decades (see, [7, 10, 11, 13, 14, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The extended horizontal linear complementarity problem is one of the most important extensions of LCP, which various authors have studied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' see [4, 6, 7] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For a given ordered set of matrices C := {C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck} ⊆ Rn×n, vector q ∈ Rn and ordered set of positive vectors d := {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk} �� Rn, the extended horizontal linear complementarity problem (for short EHLCP), denoted by EHCLP(C, d, q), is to find a vector x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk ∈ Rn such that C0x0 =q + k � i=1 Cixi, x0 ∧ x1 = 0 and (dj−xj) ∧ xj+1 = 0, 1 ≤ j ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (2) If k = 1, then EHLCP becomes the horizontal linear complementarity problem (for short HLCP), that is, C0x0 − C1x1 = q and x0 ∧ x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Further, HLCP reduces to the standard LCP by taking C0 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Due to its widespread applications in numerous domains, the horizontal linear complementarity problem has received substantial research attention from many academics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' see [13, 14, 16, 18] and reference therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Various writers have presented new classes of matrices for analysing the structure of LCP solution sets in recent years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' see for example, [1, 2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The classes of R0, P0, P, and strictly semimonotone 1 (SSM) matrices play a crucial role in the existence and uniqueness of the solution to LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For instance, P matrix (if [x ∈ Rn, x ∗ Ax ≤ 0 =⇒ x = 0]) gives a necessary and sufficient condition for the uniqueness of the solution for the LCP (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To get a similar type of existence and uniqueness results for the generalized LCPs, the notion of P matrix was extended for the set of matrices as the column W-property by Gowda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' They proved that column W- property gives the solvability and the uniqueness for the extended horizontal linear complementarity problem (EHLCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Also, they have generalized the concept of the P0-matrix as the column W0- property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Another class of matrix, the so-called SSM matrix, has importance in LCP theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This class of matrices provides a unique solution to LCP on Rn + and also gives the existence of the solution for the LCP (see, [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For a Z matrix (if all the off-diagonal entries of a matrix are non-positive), P matrix is equivalent to the SSM matrix (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' A natural question arises whether the SSM matrix can be generalized for the set of matrices in the view of EHLCP and whether we have a similar equivalence relation for the set of Z matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In this paper, we would like to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The connectedness of the solution set of LCP has a prominent role in the study of the LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say a matrix is connected if the solution set of the corresponding LCP is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In [19], Jones and Gowda addressed the connectedness of the solution set of the LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' They proved that the matrix is connected whenever the given matrix is a P0 matrix and the solution set has a bounded connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Also, they have shown that if the solution set of LCP is connected, then there is almost one solution of LCP for all q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Due to the specially structured matrices involved in the study of the connectedness of the solution to LCP, various authors studied the connectedness of LCP, see for example [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The main objectives of this paper are to answer the following questions: (Q1) In LCP theory, it is a well-known result that the R0 matrix gives boundedness to the LCP solution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The same holds true for HLCP [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This motivates the question of whether or not the notion of R0 matrix can be generalized to the set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If so, then can we expect the same kind of outcome in the EHLCP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (Q2) Given that a strictly semimonotone matrix guarantees the existence of the LCP solution and its uniqueness for q ≥ 0, it is natural to wonder whether the concept of SSM matrix can be extended to the set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If so, then whether the same result holds true for EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (Q3) Motivated by the results of Gowda and Jones [19] regarding the connectedness of the solution set of LCP, one can ask whether the solution set of EHLCP is connected if the set of matrices has the column W0 property and the solution set of the corresponding EHLCP has a bounded connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The paper’s outline is as follows: We present some basic definitions and results in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We generalize the concept of R0 matrix and prove the existence result for EHLCP in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In section 4, we introduce the SSM-W property, and we then study an existence and uniqueness result for the EHLCP when the underlying set of matrices have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In the last section, we give a necessary and sufficient condition for the connectedness of the solution set of the EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 2 Notations and Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 Notations Throughout this paper, we use the following notations: (i) The n dimensional Euclidean space with the usual inner product will be denoted by Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The set of all non-negative vectors (respectively, positive vectors) in Rn will be denoted by Rn + 2 (respectively, Rn ++ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say x ≥ 0 (respectively, > 0) if and only if x ∈ Rn + (respectively, Rn ++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) The k-ary Cartesian power of Rn will be denoted by Λ(k) n and the k-ary Cartesian power of Rn ++ will be denoted by Λ(k) n,++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The bold zero ’0’ will be used for denoting the zero vector (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ Λ(k) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) The set of all n×n real matrices will be denoted by Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We use the symbol Λ(k) n×n to denote the k-ary Cartesian product of Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iv) We use [n] to denote the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (v) Let M ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We use diag(M) to denote the vector (M11, M22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Mkk) ∈ Rn, where Mii is the iith diagonal entry of matrix M and det(M) is used to denote the determinant of matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (vi) SOL(C, d, q) will be used for denoting the set of all solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall some definitions and results from the LCP theory, which will be used frequently in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let V = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) x ∧ y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) x, y ≥ 0 and x ∗ y = 0, where ∗ is the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) x, y ≥ 0 and ⟨x, y⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 1 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then a matrix R ∈ Rn×n is column representative of C if R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j ∈ � (C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j, (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', (Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j � , ∀j ∈ [n], where R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='j is the jth column of matrix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Next, we define the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 2 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C := (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we say that C has the (i) column W-property if the determinants of all the column representative matrices of C are all positive or all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) column W0-property if there exists N := (N0, N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Nk) ∈ Λ(k+1) n×n such that C + ǫN := (C0 + ǫN0, C1 + ǫN1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck + ǫNk) has the column W-property for all ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Due to Gowda and Sznajder [4], we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n , the following are equivalent: (i) C has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) For arbitrary non-negative diagonal matrices D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk ∈ Rn×n with diag(D0 +D1 +D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0, det � C0D0 + C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) C0 is invertible and (I, C−1 0 C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', C−1 0 Ck) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 3 (iv) For all q ∈ Rn and d ∈ Λ(k−1) n,++ , EHLCP(C, d, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If k = 1 and C−1 0 exists, then HLCP(C0, C1, q) is equivalent to LCP(C−1 0 C1, C−1 0 (q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In this case, C−1 0 C1 is a P matrix if and only if for all q ∈ Rn, LCP(C−1 0 C1, C−1 0 (q)) has a unique solution (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence we have the following theorem given the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let (C0, C1) ∈ Λ(2) n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) (C0, C1) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) C0 is invertible and C−1 0 C1 is a P matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) For all q ∈ Rn, HLCP(C0, C1, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 Degree theory We now recall the definition and some properties of a degree from [2, 3] for our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let Ω be an open bounded set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose h : ¯Ω → Rn is a continuous function and a vector p /∈ h(∂Ω), where ∂Ω and ¯Ω denote the boundary and closure of Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the degree of h is defined with respect to p over Ω denoted by deg(h, Ω, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The equation h(x) = p has a solution whenever deg(h, Ω, p) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If h(x) = p has only one solution, say y in Rn, then the degree is the same overall bounded open sets containing y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This common degree is denoted by deg(h, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 Properties of the degree The following properties are used frequently here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (D1) deg(I, Ω, ·) = 1, where I is the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (D2) Homotopy invariance: Let a homotopy Φ(x, s) : Rn ×[0, 1] → Rn be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If the zero set of Φ(x, s), X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded, then for any bounded open set Ω in Rn containing the zero set X, we have deg(Φ(x, 1), Ω, 0) = deg(Φ(x, 0), Ω, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (D3) Nearness property: Assume deg(h1(x), Ω, p) is defined and h2 : ¯Ω → Rn is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If supx∈Ω∥h2(x) − h1(x)∥ < dist(p, ∂Ω), then deg(h2(x), Ω, p) is defined and equals to deg(h1(x), Ω, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following result from Facchinei and Pang [2] will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4 ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let Ω be a non-empty, bounded open subset of Rn and let Φ : ¯Ω → Rn be a continuous injective mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then deg(Φ, Ω, p) ̸= 0 for all p ∈ Φ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Note: All the degree theoretic results and concepts are also applicable over any finite dimensional Hilbert space (like Rn or Rn × Rn × Rn etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 3 R0-W property In this section, we first define the R0-W property for the set of matrices which is a natural generaliza- tion of R0 matrix in the LCP theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We then show that the R0-W property gives the boundedness of the solution set of the corresponding EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 4 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say that C has the R0-W property if the system C0x0 = k � i=1 Cixi and x0 ∧ xj = 0 ∀ j ∈ [k] has only zero solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It can be seen easily that the R0-W property coincides with R0 matrix when k = 1 and C0 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Also it is noted (see, [8]) that if k = 1, then the R0-W property referred as R0 pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To proceed further, we prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n and x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then x satisfies the following system C0x0 = q + k � i=1 Cixi and x0 ∧ xj = 0 ∀ j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As x0 ≥ 0, there exists an index set α ⊆ [n] such that (x0)i = � > 0 i ∈ α 0 i ∈ [n] \\ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since x0 ∧ x1 = 0, we have (x1)i = 0 for all i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From (d1 − x1) ∧ x2 = 0, we get (d1)i(x2)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives that (x2)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By substituting (x2)i = 0 ∀i ∈ α in (d2 − x2) ∧ x3 = 0, we obtain (x3)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Continue the process in the similar way, one can get (x4)i = (x5)i = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' = (xk)i = 0 ∀i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, x0 ∧ xj = 0 ∀ j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now prove the boundedness of the solution set of EHLCP when the involved set of matrices has the R0-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C has the R0-W property then SOL(C, d, q) is bounded for every q ∈ Rn and d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exist q ∈ Rn and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk−1) ∈ Λ(k−1) n,++ such that SOL(C, d, q) is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then there exists a sequence x(m) = (x(m) 0 , x(m) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', x(m) k ) in Λ(k+1) n such that ||x(m)|| → ∞ as m → ∞ and it satisfies C0x(m) 0 = q + k � i=1 Cix(m) i x(m) 0 ∧ x(m) 1 = 0 and (dj − x(m) j ) ∧ x(m) j+1 = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (3) From the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, equation 3 gives that C0x(m) 0 =q + k � i=1 Cix(m) i and x(m) 0 ∧ x(m) j = 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (4) As x(m) ∥x(m)∥ is a unit vector for all m, x(m) ∥x(m)∥ converges to some vector y = (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', yk) ∈ Λ(k+1) n with ||y|| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Now first divide the equation 4 by ∥x(m)∥ and then take the limit m → ∞, we get C0y0 = k � i=1 Ciyi and y0 ∧ yj = 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that y must be a zero vector as C has the R0-W property, which contradicts the fact that ||y|| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore SOL(C, d, q) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 Degree of EHLCP Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., dk−1) ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We define a function F : Λ(k+1) n → Λ(k+1) n as F(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (5) We denote the degree of F with respect to 0 over bounded open set Ω ⊆ Λ(k+1) n as deg(C, Ω, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is noted that if C has the R0-W property, in view of the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, F(x) = 0 ⇔ x = 0 which implies that deg(C, Ω, 0) = deg(C, 0) for any bounded open set Ω contains the origin in Λ(k+1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We call this degree as EHLCP-degree of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now prove an existence result for EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose the following hold: (i) C has the R0-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) deg(C, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then EHLCP(C, d, q) has non-empty compact solution for all q ∈ Rn and d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As the solution set of EHLCP is closed, it is enough to prove that the solution set is non-empty and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We first define a homotopy Φ : Λ(k+1) n × [0, 1] → Λ(k+1) n as Φ(x, s) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi − sq x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, Φ(x, 0) = F(x) and Φ(x, 1) = F(x) − ˆq, where ˆq = (q, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='0) ∈ Λ(k+1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By using the similar argument as in above Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2, we can easily show that the zero set of homotopy, X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From the property of degree (D2), we get deg(F, Ω, 0) = deg(F − ˆq, Ω, 0) for any open bounded set Ω containing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As deg(F, Ω, 0) = deg(C, 0) ̸= 0, we obtain deg(F − ˆq, Ω, 0) ̸= 0 which implies SOL(C, d, q) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the R0-W property, by Theorem 3, SOL(C, d, q) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 4 SSM-W property In this section, we first define the SSM-W property for the set of matrices which is a generalization of the SSM matrix in the LCP theory, and we then prove that the existence and uniqueness result for the EHLCP when the involved set of matrices have the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall that an n × n real matrix M is called strictly semimonotone (SSM) matrix if [x ∈ Rn +, x ∗ Mx ≤ 0 ⇒ x = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We generalize this concept to the set of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 6 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say that C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property if {C0x0 = k � i=1 Cixi, xi ≥ 0 and x0 ∗ xi ≤ 0 ∀i ∈ [k]} ⇒ x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., xk) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C has the SSM-W property, then the followings hold: (i) C−1 0 exists and C−1 0 Ci is a strict semimonotone matrix for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) (I, C−1 0 C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', C−1 0 Ck) has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii) (P T C0P, P T C1P, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', P T CkP) has the SSM-W property for any permutation matrix P of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i): Suppose there exists a vector x0 ∈ Rn such that C0x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we have C0x0 = C10 + C20 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ck0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives that x0 = 0 as C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus C0 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Now we prove the second part of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Without loss of generality, it is enough to prove that C−1 0 C1 is a strictly semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exists a vector y ∈ Rn such that y ≥ 0 and y ∗ (C−1 0 C1)y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let y0 := (C−1 0 C1)y, y1 := y and yi := 0 for all 2 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we get C0y0 = C1y1 + C2y2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ciyi + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='. + Ckyk, yj ≥ 0 and y0 ∗ yj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since C has the SSM-W property, yj = 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus C−1 0 C1 is a strict semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii): It follows from the definition of the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (iii): Let x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n such that (P T C0P)x0 = k � i=1 (P T CiP)xi, xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As P is a non-negative matrix and PP T = P T P, we can rewrite the above equation as C0Px0 = k � i=1 CiPxi, Pxj ≥ 0 and Px0 ∗ Pxj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By the SSM-W property of C, Pxj = 0 for all 0 ≤ j ≤ k which implies x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In the above Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, it can be seen easily that the converse of the item (ii) and (iii) are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' But the converse of item (i) need not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following example illustrates this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, C2) ∈ Λ(3) 2×2, where C0 = � 1 0 0 1 � , C1 = � 1 −2 0 1 � , C2 = � 1 0 −2 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is easy to check that C−1 0 C1 = C1 and C−1 0 C2 = C2 are P matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, C−1 0 C1 and C−1 0 C2 are SSM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let x = (x0, x1, x2) = ((0, 0)T , (1, 1)T , (1, 1)T ) ∈ Λ(3) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we can see that the non-zero x satisfies C0x0 = C1x1 + C2x2, x1 ≥ 0, x2 ≥ 0 and x0 ∗ x1 = 0 = x0 ∗ x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So C can not have the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 7 The following result is a generalization of a well-known result in matrix theory that every P matrix is a SSM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C has the column W-property, then C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exists a non-zero vector x = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n such that C0x0 = k � i=1 Cixi, xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Consider a vector y ∈ Rn whose jth component is given by yj = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −1 if (x0)j > 0 1 if (x0)j < 0 1 if (x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k] 0 if (x0)j = 0 and (xi)j = 0 for all i ∈ [k] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As x is a non-zero vector, y must be a non-zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Consider the diagonal matrices D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk which are defined by (D0)jj = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (x0)j if (x0)j > 0 −(x0)j if (x0)j < 0 0 if(x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k] 1 if (x0)j = 0 and (xi)j = 0 for all i ∈ [k] and for all i ∈ [k], (Di)jj = � 0 if (x0)j > 0 (xi)j else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is easy to verify that D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk are non-negative diagonal matrices and diag(D0 + D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' And also note that x0 = −D0y and xi = Diy ∀i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (6) By substituting the Equation 6 in C0x0 = �k i=1 Cixi, we get C0(−D0y) = k � i=1 CiDi(y) ⇒ � C0D0 + C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that det(C0D0 + C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, C does not have the column W-property from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore, C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following example illustrates that the converse of the above theorem is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, C2) ∈ Λ(3) 2×2 such that C0 = �1 0 0 1 � , C1 = �1 1 1 1 � , C2 = �1 1 1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose w = (x, y, z) ∈ Λ3 2 such that C0x = C1y + C2z and y, z ≥ 0, x ∗ y ≤ 0, x ∗ z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 8 From C0x = C1y + C2z, we get �x1 x2 � = �y1 + y2 + z1 + z2 y1 + y2 + z1 + z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As x ∗ y ≤ 0, x ∗ z ≤ 0 and from the above equation, we have y1(y1 + y2 + z1 + z2) ≤ 0 and y2(y1 + y2 + z1 + z2) ≤ 0, z1(y1 + y2 + z1 + z2) ≤ 0 and z2(y1 + y2 + z1 + z2) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (7) Since y, z ≥ 0, from the equation 7, we get x = y = z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As det(C1) = 0, by the definition of the column W-property, C does not have the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now give a characterization for SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property if and only if (C0, C1D1+ C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) ∈ Λ(2) n×n has the SSM-W property for any set of non-negative diagonal matrix (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk) ∈ Λ(k) n×n with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Necessary part: Let (D1, D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk) ∈ Λ(k) n×n be the set of non-negative diagonal matrix with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose there exist vectors x0 ∈ Rn and y ∈ Rn + such that C0x0 = � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � y and x0 ∗ y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For each i ∈ [k], we set xi := Diy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As each Di is a non-negative diagonal matrix, from x0 ∗ y ≤ 0, we get x0 ∗ xi ≤ 0 ∀i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then we have C0x0 = C1x1 + C2x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ckxk, xi ≥ 0, x0 ∗ xi ≤ 0 ∀i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property of C, we must have x0 = x1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' = xk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies x1 + x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + xk = (D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk)y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='. + Dk) > 0, we have y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the necessary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Sufficiency part: Let x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n such that C0x0 = C1x1 + C2x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ckxk and xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (8) We now consider an n × k matrix X whose jth column as xj for j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, X = [x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' xk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let S := {i ∈ [k] : ith row sum of X is zero}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From this, we define a vector y ∈ Rn and diagonal matrices D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., Dk such that yi = � 1 i /∈ S 0 i ∈ S and (Dj)ii = � (xj)i i /∈ S 1 i ∈ S , where (Dj)ii is the diagonal entry of Dj for all j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It can be seen easily that Djy = xj for all j ∈ [k] and each Dj is a non-negative diagonal matrix with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='+ Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore, from equation 8, we get C0x0 = � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � y, x0 ∗ y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From the hypothesis, we get x0 = 0 = y which implies x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the sufficiency part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 9 We now give a characterization for the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the column-W property if and only if (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) ∈ Λ(2) n×n has the column-W property for any set of non-negative diagonal matrices D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk of order n with diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Necessary part: It is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Sufficiency part: Let {E0, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ek} be a set of non-negative diagonal matrices of order n such that diag(E0 + E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ek) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We claim that det(C0E0 + C1E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkEk) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To prove this, we first construct a set of non-negative diagonal matrices D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk and E as follows: (Dj)ii = � Ej ii if �k m=1 Em ii ̸= 0 1 if �k m=1 Em ii = 0 and Eii = � 1 if �k m=1 Em ii ̸= 0 0 if �k m=1 Em ii = 0 , where (Dj)ii is iith diagonal entry of Dj for j ∈ [k] and Eii is iith diagonal entry of matrix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By an easy computation, we have DjE = Ej ∀j ∈ [k] and diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From diag(E0 + E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Ek) > 0, we get diag(E0 + E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As DjE = Ej ∀j ∈ [k] and (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has column W-property, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2, we have det(C0E0 + C1E1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkEk) = det(C0E0 + C1D1E + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDkE) = det(C0E0 + (C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk)E) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence C has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' A well-known result in the standard LCP is that strictly semimonotone matrix and P matrix are equivalent in the class of Z matrices (see, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Analogue this result, we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n such that C−1 0 Ci be a Z matrix for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) C has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) =⇒ (ii): It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) =⇒ (i): Let {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Dk} be the set of non-negative diagonal matrices of order n such that diag(D1 + D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + Dk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' In view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='6, it is enough to prove that (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='5, we have (C0, C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, � I, C−1 0 � C1D1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk �� has the SSM-W property and C−1 0 � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � is a strict semimonotone matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C−1 0 Ci is a Z matrix, we get C−1 0 � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � is also a Z matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence C−1 0 � C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk � is a P matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3, (C0, C1D1 + C2D2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' + CkDk) has the column W-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence we have our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λk+1 n×n such that C−1 0 Ci be a Z matrix for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) For all q ∈ Rn and d ∈ Λ(k−1) n,++ , EHLCP(C, d, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (i) =⇒ (ii): It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (ii) =⇒ (i): It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 10 In the standard LCP [3], the strictly semimonotone matrix gives the existence of a solution of LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now prove that the same result holds in EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property, then SOL(C, d, q) ̸= ∅ for all q ∈ Rn and d ∈ Λ(k+1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property, C has the R0-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, it is enough to prove that deg(C, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' To prove this, we consider a homotopy Φ : Λ(k+1) n × [0, 1] → Λ(k+1) n as Φ(x, t) = t \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 x1 x2 x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + (1 − t) \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let F(x) := Φ(x, 0) and G(x) := Φ(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We first prove that the zero set X = {x : Φ(x, t) = 0 for some t ∈ [0, 1]} of homotopy Φ contains only zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 1: Suppose t = 0 or t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If t = 0, then Φ(x, 0) = 0 =⇒ F(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C has the SSM-W property, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, we have F(x) = 0 ⇒ x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If t = 1, then Φ(x, 1) = 0 =⇒ G(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Again by C has the SSM-W property, C−1 0 exists, which implies that G is a one-one map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, G(x) = 0 ⇒ x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 2: Suppose t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then Φ(x, t) = 0 which gives that \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi x0 ∧ x1 (d1 − x1) ∧ x2 (d2 − x2) ∧ x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = −α \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 x1 x2 x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , where α = t 1 − t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (9) From the second row of above equation, we have x0 ∧ x1 = −αx1 =⇒ min{x0 + αx1, (1 + α)x1} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, we get x1 ≥ 0 and (x0 + αx1) ∗ (1 + α)x1 = 0 which implies that x0 ∗ x1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Set ∆ := {i ∈ [n] : (x1)i > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, we have (x0)i = � ≤ 0 if i ∈ ∆ ≥ 0 if i /∈ ∆ and (x1)i = � > 0 if i ∈ ∆ = 0 if i /∈ ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (10) From third row of the equation 9, we have (d1 − x1) ∧ x2 = −αx2 which is equivalent min{d1 − x1 + αx2, (1 + α)x2} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives that x2 ≥ 0 and (d1 − x1 + αx2) ∗ (1 + α)x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As d1 > 0 and from the last term in equation 10, we have (x2)i = � ≥ 0 if i ∈ ∆ = 0 if i /∈ ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 11 This leads that x0 ∗ x2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By continuing the similar argument for the remaining rows, we get xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From the first row of the equation 9, the vectors x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) satisfies C0(1 + α)x0 = k � i=1 Cixi and xj ≥ 0, x0 ∗ xj ≤ 0, j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, x = 0 as C has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From both cases, we get X contains only zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By the homotopy invariance property of degree (D2), we have deg(Φ(x, 0), Ω, 0) = deg � Φ(x, 1), Ω, 0 � for any bounded open set containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As G is a continuous one-one function, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4, we have deg � C, 0 � = deg � Φ(x, 0), Ω, 0 � = deg � F, Ω, 0 � = deg � G, Ω, 0 � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall that a matrix A ∈ Rn×n is said to be a M matrix if it is Z matrix and A−1(Rn +) ⊆ Rn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We prove a uniqueness result for EHLCP when q ≥ 0 and d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the SSM-W property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C0 is a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' then for every q ∈ Rn + and for every d ∈ Λ(k−1) n,++ , EHLCP(C, d, q) has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let q ∈ Rn + and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk−1) ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We first show (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C0 is a M matrix and q ∈ Rn +, we have C−1 0 q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If we set y = (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', yk) := (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ Λ(k+1) n , then we can see easily that (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', yk) satisfies that C0y0 = q + k � i=1 Ciyi, y0 ∧ y1 = 0 and (dj − yj) ∧ yj+1 = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) ∈ Λ(k+1) n is an another solution to EHLCP(C, q, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, C0x0 = q + k � i=1 Cixi, x0 ∧ x1 = 0, (dj − xj) ∧ xj+1 = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (11) From the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, we have C0x0 = q + k � i=1 Cixi and x0 ∧ xj = 0 ∀ j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (12) We let z := x − y, then z = (x0 − C−1 0 q, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By an easy computation, from Equation 12, we get C0(x0 − C−1 0 q) = k � i=1 Cixi and xj ≥ 0, (x0 − C−1 0 q) ∗ xj = x0 ∗ xj − C−1 0 q ∗ xj = −C−1 0 q ∗ xj ≤ 0 ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since C has the SSM-W property, z = 0 which implies that (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', xk) = (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 12 5 Connected solution set and Column W0 property In this section, we give a necessary and sufficient condition for the connected solution set of the EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We say that C is connected if SOL(C, d, q) is connected for all q ∈ Rn and for all d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now recall some definitions and results to proceed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' [22] A subset of Rn is said to be a semi-algebraic set it can be represented as, S = s� u=1 ru � v=1 {x ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' fu,v(x) ∗uv 0}, where for all u ∈ [s] and for all v ∈ [ru], ∗uv ∈ { >, =} and fu,v is in the space of all real polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1 ([22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let S be a semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then S is connected iff S is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The SOL(C, d, q) is a semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It is clear from the definition of SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following result gives a necessary condition for a connected solution whenever C0 is a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C0 ∈ Rn×n be a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If C = (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n is connected, then SOL(C, d, q) = {(C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0)} for all q ∈ Rn ++ and for all d ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let q ∈ Rn ++ and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', dk−1) ∈ Λ(k−1) n,++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It can be seen from the proof of The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='10 that x = (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) ∈ SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now show that x is the only solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Assume contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose y is another solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As SOL(C, d, q) is con- nected, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='1, it is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, there exists a path γ = (γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', γk) : [0, 1] → SOL(C, d, q) such that γ(0) = x, γ(1) = y and γ(t) ̸= x ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let {tm} ⊆ (0, 1) be a sequence such that tm → 0 as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then, by the continuity of γ, γ(tm) → γ(0) = x as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since � γ0(tm), γ1(tm), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='γk(tm) � ∈ SOL(C, d, q), C0γ0(tm) = q + k � i=1 Ciγi(tm), γ0(tm) ∧ γ1(tm) = 0 and � dj − γj(tm) � ∧ γ(j+1)(tm) = 0 ∀j ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Now we claim that there exists a subsequence {tml} of {tm} such that � γj(tml) � i ̸= 0, for some j ∈ [k] and for some i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose the claim is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This means that for given any subsequence {tml} of {tm}, there exists m0 ∈ N such that for all ml ≥ m0, we have � γj(tml) � i = 0 ∀i ∈ [n] ∀j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 13 So, γj(tm) is an eventually zero sequence for all j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that there exists a natural number m0 such that γ1(tm) = γ2(tm) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' = γk(tm) = 0 ∀m ≥ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As � γ0(tm), γ1(tm), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='γk(tm) � ∈ SOL(C, d, q), we get γ0(tm) = C−1 0 (q) ∀m ≥ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This gives us that γ(tm) = x for all m ≥ m0 which contradicts the fact that γ(tm) ̸= x for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore, our claim is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' No loss of generality, we assume a sequence {tm} itself satisfies the condition � γj(tm) � i ̸= 0, for some j ∈ [k] and for some i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' We now consider the following cases for possibilities of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 1 : If j = 1, then (γ0(tm))i(γ1(tm))i = 0 which leads to (γ0(tm))i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that 0 = lim m→∞ γ0(tm)i = (C−1 0 q)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' But (C−1 0 q) > 0 as C0 is a M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, j ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Case 2 : If 2 ≤ j ≤ k, then we have (dj−1 − γj−1(tm))i(γj(tm))i = 0 which gives that (dj−1 − γj−1(tm))i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By taking limit m → ∞, 0 = lim m→∞(dj−1 − γj−1(tm))i = (dj−1)i − (γj−1(0))i = (dj−1)i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' From both cases, there is no such a j exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This contradicts the fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence x = (C−1 0 q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', 0) is the only solution to EHLCP(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' The following result gives a sufficient condition for a connected solution to EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let C := (C0, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck) ∈ Λ(k+1) n×n has the column W0-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If SOL(C, d, q) has a bounded connected component, then SOL(C, d, q) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If SOL(C, d, q) = ∅, then we have nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let SOL(C, d, q) ̸= ∅ and A be a con- nected component of SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' If SOL(C, d, q) = A, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Suppose SOL(C, d, q) ̸= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Then there exists y = (y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='., yk) ∈ SOL(C, d, q)\\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As A is a bounded connected component of SOL(C, d, q), we can find an open bounded set Ω ⊆ Λ(k+1) n which contains A and it does not intersect with other component of SOL(C, d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Therefore y /∈ Ω and ∂(Ω) ∩ SOL(C, d, q) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since C has the column W0-property, there exists N := (N0, N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Nk) ∈ Λ(k+1) n×n such that C + ǫN := (C0 + ǫN0, C1 + ǫN1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', Ck + ǫNk) has the column W-property for every ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Let z = (z0, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=', zk) ∈ A and ǫ > 0, we define functions H1, H2 and H3 as follows: H1(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 C0x0 − �k i=1 Cixi − q x0 ∧ x1 (d1 − x1) ∧ x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , H2(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 (C0 + ǫN0)x0 − �k i=1(Ci + ǫNi)xi + (�k i=1 ǫNiyi − ǫN0y0 − q) x0 ∧ x1 (d1 − x1) ∧ x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , 14 H3(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 (C0 + ǫN0)x0 − �k i=1(Ci + ǫNi)xi + (�k i=1 ǫNizi − ǫN0z0 − q) x0 ∧ x1 (d1 − x1) ∧ x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (dk−1 − xk−1) ∧ xk \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' By putting x = y in H2(x), and x = z in H1(x) and H3(x), we get H1(z) = H2(y) = H3(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' For ǫ is near to zero, deg(H1, Ω, 0)= deg(H2, Ω, 0)= deg(H3, Ω, 0) due to the nearness property of degree (D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As z ∈ Ω is a solution to H3(x) = 0 and C + ǫN has the column W-property, we get deg(H3, Ω, 0) ̸= 0 by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Since deg(H2, Ω, 0)= deg(H3, Ω, 0), we have deg(H2, Ω, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' This implies that if we set q2 := q+ǫN0y0−�k i=1 ǫNiyi, then EHLCP(C + ǫN, d, q2) must have a solution in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' As C + ǫN has the column W-property, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='2, EHLCP(C + ǫN, d, q2) has a unique solution which must be equal to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' So, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' It gives us a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Hence SOL(C, d, q) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Thus SOL(C, d, q) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 6 Conclusion In this paper, we introduced the R0-W property and SSM-W properties and then studied the existence and uniqueness result for EHLCP when the underlying set of matrices has these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Last, we gave a necessary and sufficient condition for the connectedness of the solution set of the EHLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Declaration of Competing Interest The authors have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Acknowledgements The first author is a CSIR-SRF fellow, and he wants to thank the Council of Scientific & Industrial Research(CSIR) for the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Cottle, J.' metadata={'source': 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Algorithms in Real Algebraic Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' Berlin: SpringerVerlag;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'} +page_content=' (2006) 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfg_zf/content/2301.01479v1.pdf'}