An output-sensitive algorithm for multi-parametric LCPs with sufficient matrices

TL;DR

This paper presents an output-sensitive algorithm for solving multi-parametric linear complementarity problems with sufficient matrices, providing a polyhedral decomposition and a piecewise affine solution mapping.

Contribution

The paper introduces a novel output-sensitive algorithm for multi-parametric LCPs with sufficient matrices, including a method to handle degeneracy using lexicographic perturbation.

Findings

Algorithm's complexity is polynomial in input size and linear in output size for non-degenerate cases.

Provides a lexicographic perturbation technique to resolve degeneracy.

Demonstrates the algorithm's effectiveness in constructing solution mappings.

Abstract

This paper considers the multi-parametric linear complementarity problem (pLCP) with sufficient matrices. The main result is an algorithm to find a polyhedral decomposition of the set of feasible parameters and to construct a piecewise affine function that maps each feasible parameter to a solution of the associated LCP in such a way that the function is affine over each cell of the decomposition. The algorithm is output-sensive in the sense that its time complexity is polynomial in the size of the input and linear in the size of the output, when the problem is non-degenerate. We give a lexicographic perturbation technique to resolve degeneracy as well. Unlike for the non-parametric case, the resolution turns out to be nontrivial, and in particular, it involves linear programming (LP) duality and multi-objective LP.