# A New Parameterization Algorithm for the Linear Complementarity Problem

## Abstract

We study a new parameterization algorithm for the *P*-matrix restriction of the linear complementarity problem. This parameterization can be considered similar to a path-following penalty method, and introduces to this class of algorithms a primal-dual symmetry similar to that seen in the highly successful path-following logarithmic barrier methods. The trajectory associated with the parameterization is distinguished by a naturally defined starting point and by a piecewise characterization as a fractional polynomial function of a single parameter. The trajectory can be followed exactly using root isolation and a simplex-like pivoting scheme. Convergence is guaranteed for a weakly-regular subclass of *P*-matrix LCPs. We show that under a weakly-regular and sign-invariant distribution for the input matrices and vectors, the average number of pieces in the trajectory is *O*(*n* ^{2}), where *n* is the dimension of the problem space, and hence that the path-following algorithm has average-case polynomial running time.

## Keywords

Linear complementarity path following probabilistic analysis sign-invariant.## Preview

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