Γ-robust linear complementarity problems

Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems; e.g., in game-theoretic settings like the bimatrix game or in energy market models like for electricity or natural gas. In all of these situations, rational choice represented by formal optimization principles is almost always due to incomplete information about the optimization problem's data in practice. However, while optimization under such uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty is still in its infancy. Being more specific, although there is a reasonable amount of literature on the stochastic treatment of uncertain complementarity problems, only very little is known about robust techniques for these problems. In this paper, we extend the theory of strictly robust linear complementarity problems (LCPs) to Γ-robust settings in the sense of Bertsimas and Sim. As in the strictly robust case, there is almost no hope for existence of worst-case-hedged equilibria. Thus, we study the minimization of the worst-case gap function of Γ-robust counterparts of LCPs. For box and l1-norm uncertainty sets we derive tractable convex counterparts for monotone LCPs and study their feasibility as well as the existence and uniqueness of their solutions. To this end, we consider both the situations of uncertainty in the LCP vector q and in the LCP matrix M. We additionally study so-called ρ-robust solutions, i.e., solutions of relaxed uncertain LCPs. Finally, we illustrate the Γ-robust concept applied to LCPs in the light of the above mentioned classical examples of bimatrix games and market equilibrium modeling.