Karush–Kuhn–Tucker conditions and duality for a class of convex adjustable robust optimization problem

This paper investigates optimality conditions and duality for a class of convex adjustable robust optimization problem (ARP). We first propose some constraint qualifications (CQs): the Abadie CQ, the local Farkas–Minkowski CQ, Mangasarian-Fromovitz CQ, and Slater CQ; and give some relationships between these CQs. Then we employ them to derive necessary and sufficient optimality conditions for the optimal solution of (ARP). These conditions are form of Karush–Kuhn–Tucker multiplier rules. We also introduce the Wolfe and Mond-Weir duality schemes and discuss weak, strong, and converse duality results. As an application, some optimality conditions for robust optimization problem are obtained. We include many examples for analyzing and illustrating our results.