On robust solutions to uncertain linear complementarity problems and their variants

TL;DR

This paper introduces a distribution-free, convex programming approach for obtaining robust solutions to uncertain linear complementarity problems, including non-monotone and hierarchical variants, with promising preliminary numerical results.

Contribution

It develops a tractable convex programming framework for robust solutions to uncertain LCPs under specified uncertainty sets, extending to non-monotone and hierarchical problems.

Findings

Robust solutions can be obtained via convex programs under certain assumptions.

Non-monotone problems lead to low-dimensional nonconvex quadratically constrained quadratic programs.

Preliminary numerics indicate the approach's potential for practical applications.

Abstract

A popular approach for addressing uncertainty in variational inequality problems is by solving the expected residual minimization (ERM) problem. This avenue necessitates distributional information associated with the uncertainty and requires minimizing nonconvex expectation-valued functions. We consider a distinctly different approach in the context of uncertain linear complementarity problems with a view towards obtaining robust solutions. Specifically, we define a robust solution to a complementarity problem as one that minimizes the worst-case of the gap function. In what we believe is amongst the first efforts to comprehensively address such problems in a distribution-free environment, we show that under specified assumptions on the uncertainty sets, the robust solutions to uncertain monotone linear complementarity problem can be tractably obtained through the solution of a single convex program. We also define uncertainty sets that ensure that robust solutions to non-monotone generalizations can also be obtained by solving convex programs. More generally, robust counterparts of uncertain non-monotone LCPs are proven to be low-dimensional nonconvex quadratically constrained quadratic programs. We show that these problems may be globally resolved by customizing an existing branching scheme. We further extend the tractability results to include uncertain affine variational inequality problems defined over uncertain polyhedral sets as well as to hierarchical regimes captured by mathematical programs with uncertain complementarity constraints. Preliminary numerics on uncertain linear complementarity and traffic equilibrium problems suggest that the presented avenues hold promise.