Pareto Efficiency in Nonlinear Robust Optimization

Pareto Robustly Optimal (PRO) solutions were introduced by Iancuand Trichakis for linear robust optimization problems with compact polyhe-dral uncertainty sets. In this paper, we investigate PRO solutions for nonlinearrobust optimization problems with a convex uncertainty set. Precisely, we con-sider a nonlinear robust optimization problem maxx∈Xminp∈Uf(p,x), where X is thefeasible set and U is the uncertainty set. We assume that f is linear with re-spect to the uncertain parameter p. The feasible set X and the uncertain setU are assumed to be nonempty and convex. We provide a full characterizationof these solutions under some assumptions. Additionally, we present a methodcapable of producing PRO solutions for the considered nonlinear robust opti-mization problems. Furthermore, we discuss the conditions under which PROsolutions coincide with Robust Optimal solutions. Given the important roleof interior points in the calculation and characterization of PRO solutions,we offer a theoretical characterization of the relative interior points of a gen-eral convex (possibly non-polyhedral) set, and finally present two continuousoptimization problems that yield relative interior points. In addition to theirtheoretical appeal, our results extend the applicability of PRO solutions to awide range of problems and open new paths for future research in this area. Mathematics Subject Classification (2000) 90C17 · 90C30 · 90C29