A Structure-Preserving Pivotal Method for Affine Variational Inequalities

TL;DR

This paper introduces PATHAVI, a structure-preserving pivotal method for efficiently solving large-scale affine variational inequalities with theoretical guarantees and high accuracy, without problem reduction.

Contribution

PATHAVI is a novel, structure-preserving solver that handles large-scale AVI problems efficiently while maintaining theoretical properties and extending problem applicability.

Findings

PATHAVI outperforms existing methods in large-scale AVI problems.

It maintains structure and theoretical guarantees without problem reduction.

Effective in applications like friction contact and Nash Equilibria.

Abstract

Affine variational inequalities (AVI) are an important problem class that generalize systems of linear equations, linear complementarity problems and optimality conditions for quadratic programs. This paper describes PATHAVI, a structure-preserving pivotal approach, that can process (solve or determine infeasible) large-scale sparse instances of the problem efficiently, with theoretical guarantees and at high accuracy. PATHAVI implements a strategy that is known to process models with good theoretical properties without reducing the problem to specialized forms, since such reductions may destroy structure in the models and can lead to very long computational times. We demonstrate formally that PATHAVI implicitly follows the theoretically sound iteration paths, and can be implemented in a large scale setting using existing sparse linear algebra and linear programming techniques without employing a reduction. We also extend the class of problems that PATHAVI can process. The paper demonstrates the effectiveness of our approach by comparison to the PATH solver used on a complementarity reformulation of the AVI in the context of applications in friction contact and Nash Equilibria problems. PATHAVI is a general purpose solver, and freely available under the same conditions as PATH .