# An extension of Lemke’s method for the solution of a generalized Linear Complementarity Problem

• Joaquim J. Júdice
• Ana M. Faustino
II Mathematical Programming II.3 Linear Programming And Complementarity
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)

## Abstract

In this paper an extension of Lemke’s method for the solution of the Linear Complementarity Problem with Bounds (BLCP) is presented. This problem consists of finding vectors z∈Rn and w∈Rn such that
$$\begin{gathered}w = q + Mz \hfill \\a_i \leqslant z_i \leqslant b_i \hfill \\z_i = a_i \Rightarrow w_i \geqslant 0 i = 1,...,n \hfill \\z_i = b_i \Rightarrow w_i \leqslant 0 \hfill \\a_i < z_i < b_i \Rightarrow w_i = 0 \hfill \\\end{gathered}$$
where q∈Rn and M∈Rn×n are given and −∞≤ai<bi≤+∞ for all i=1,..., n. It is shown that the algorithm can process the BLCP when M is a positive semi-definite matrix or when all the bounds ai and bi are finite. The performance of the algorithm for the solution of large-scale BLCPs is briefly discussed.

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© International Federation for Information Processing 1992

## Authors and Affiliations

• Joaquim J. Júdice
• 1