Book•
Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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Dissertation•
21 Jan 2011
3 citations
Cites background from "Linear complementarity, linear and ..."
...Hence, it follows from Theorem 3.15 in Murty [72] (page 213) that (wk, zk) in (19) − (21) are uniquely determined for (q, G)....
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...15 in Murty [72] (page 213) that (wk, zk) in (19) − (21) are uniquely determined for (q, G)....
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TL;DR: This paper introduces total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem and shows that if the LCP is totally dual integral, then all basic solutions are integral.
Abstract: In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.
3 citations
3 citations
01 Jan 2004
TL;DR: In this paper, the linear complementarity problem is converted to a system of semismooth nonlinear equations by using smoothing technique and then the Levenberg-Marquardt type method is used to solve this system.
Abstract: In this paper, we convert the linear complementarity problem to a system of semismooth nonlinear equations by using smoothing technique. Then we use Levenberg-Marquardt type method to solve this system. Taking advantage of the new results obtained by Dan, Yamashita and Fukushima [11, 33], the global and local superlinear convergence properties of the method are obtained under very mild conditions. Especially, the algorithm is locally superlinearly convergent under the assumption of either strict complementarity or certain nonsingularity. Preliminary numerical experiments are reported to show the efficiency of the algorithm.
3 citations