Existence of solutions to $$\Gamma $$ -robust counterparts of gap function formulations of uncertain LCPs with ellipsoidal uncertainty sets

Abstract

In this paper, we give some existence theorems of solutions to \(\Gamma \)-robust counterparts of gap function formulations of uncertain linear complementarity problems, in which \(\Gamma \) plays a role in adjusting the robustness of the model against the level of conservatism of solutions. If the \(\Gamma \)-robust uncertainty set is nonconvex, it is hard to prove the existence of solutions to the corresponding robust counterpart. Using techniques of asymptotic functions, we establish existence theorems of solutions to the corresponding robust counterpart. For the case of nonconvex \(\Gamma \)-robust ellipsoidal uncertainty sets, these existence results are not proved in the paper [Krebs et al., Int. Trans. Oper. Res. 29 (2022), pp. 417–441]; for the case of convex \(\Gamma \)-robust ellipsoidal uncertainty sets, our existence theorems are obtained under the conditions which are much weaker than those in Krebs’ paper. Finally, a case study for the uncertain traffic equilibrium problem is considered to illustrate the effects of nonconvex uncertainty sets on the level of conservatism of robust solutions.