On the Lipschitz continuity of the solution map in linear complementarity problems over second-order cone

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• 作者：R. Balaji ; ^{balaji5@iitm.ac.in" class="auth_mail" title="E-mail the corresponding author} ; K. Palpandi
• 关键词：90C33 ; 65K05
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：510
• 期：Complete
• 页码：146-159
• 全文大小：324 K

文摘

Let $\mathcal{K}\subseteq {\mathit{I}\mathit{R}}^n$ denote the second-order cone. Given an n×n$n\times n$ real matrix M   and a vector $q\in {\mathit{I}\mathit{R}}^n$, the second-order cone linear complementarity problem SOLCP(M,q)$\text{SOLCP}(M,q)$ is to find a vector c33d61887b905255be2d016de5a780fc">$x\in {\mathit{I}\mathit{R}}^n$ such that

We say that 20cea0b08ace2edbb41177c3316" title="Click to view the MathML source">M∈Q$M\in \mathbf{Q}$ if SOLCP(M,q)$\text{SOLCP}(M,q)$ has a solution for all $q\in {\mathit{I}\mathit{R}}^n$. An n×n$n\times n$ real matrix A is said to be a Z-matrix with respect to K$\mathcal{K}$ iff:

Let Φ_M(q)${\mathrm{\Phi }}_M(q)$ denote the set of all solutions to SOLCP(M,q)$\text{SOLCP}(M,q)$. The following results are shown in this paper:

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If M∈Z∩Q$M\in \mathbf{Z}\cap \mathbf{Q}$, then Φ_M${\mathrm{\Phi }}_M$ is Lipschitz continuous if and only if M   is positive definite on the boundary of K$\mathcal{K}$.

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If M   is symmetric, then Φ_M${\mathrm{\Phi }}_M$ is Lipschitz continuous if and only if M is positive definite.