On Robust Solutions to Uncertain Linear Complementarity Problems and their Variants

A popular approach for addressing uncertainty in variational inequality problems requires solving the expected residual minimization problem [X. Chen and M. Fukushima, Math. Oper. Res., 30 (2005), pp. 1022--1038, X. Chen, R. J.-B. Wets, and Y. Zhang, SIAM J. Optim., 22 (2012), pp. 649--673]. This avenue necessitates distributional information associated with the uncertainty and requires minimizing a suitably defined nonconvex expectation-valued function. Alternatively, we consider a distinctly different approach in the context of uncertain linear complementarity problems (LCPs) with a view toward 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 among the first efforts to comprehensively address such problems in a distribution-free environment, under prescribed assumptions on the uncertainty sets, the robust solutions to the uncertain monotone LCP can be tractably obtained throu...