Γ‐robust linear complementarity problems with ellipsoidal uncertainty sets

Abstract

We study uncertain linear complementarity problems (LCPs), that is, problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Γ‐robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon Krebs and Schmidt (2020). There, we studied Γ‐robustified LCPs for ℓ1‐ and box‐uncertainty sets, whereas we now focus on ellipsoidal uncertainty sets. For uncertainty in q or M, we derive conditions for the tractability of the robust counterparts. For these counterparts, we also give conditions for the existence and uniqueness of their solutions. Finally, a case study for the uncertain traffic equilibrium problem is considered, which illustrates the effects of the values of Γ on the feasibility and quality of the respective robustified solutions.

Affinely Adjustable Robust Linear Complementarity Problems

“…This approach is discussed in [28,29]. The Γ-robust approach is discussed in [19,20]. The main conceptual problem with strictly as well as Γ-robust LCPs is that one usually cannot prove the existence of a solution.…”

“…Consequently, several less conservative notions of robustness have been developed during the last twenty years; see, e.g., [5,6,25] for Γ-robustness, [13] for light robustness, [2,3,30] for adjustable robustness, or [1] for deciding robustness in a fully adjustable setting with an empty first stage. Following the idea of studying less conservative notions of robustness, the concept of Γ-robustness has been applied to LCPs in [20] for the case of ℓ 1 -and box-uncertainty sets and in [19] for the case of ellipsoidal uncertainties. Applications of Γ-robust LCPs in the area of power markets or traffic equilibrium problems can be found in [8,18,19].…”

“…Following the idea of studying less conservative notions of robustness, the concept of Γ-robustness has been applied to LCPs in [20] for the case of ℓ 1 -and box-uncertainty sets and in [19] for the case of ellipsoidal uncertainties. Applications of Γ-robust LCPs in the area of power markets or traffic equilibrium problems can be found in [8,18,19]. To the best of our knowledge, the given and rather short list of papers on robust LCPs is complete.…”

“…Unfortunately, almost all the papers on robust LCPs cited above make the observation that strong characterizations and, thus, existence of robust solutions to LCPs cannot be ensured because the requirement that a point is a complementarity solution for all realizations of uncertainty is very strong. This observation is made in [28] for strict robustness and in [19,20] for Γ-robustness. As a remedy, the authors study the LCP's quadratic gap function and consider the existence and uniqueness of solutions or the tractability of problems in which the complementarity condition is not strictly demanded but in which its violation is penalized in the LCP's gap function.…”