Robust Duality for Multi-dimensional Variational Control Problem with Data Uncertainty

Abstract

Duality is a powerful tool to study optimization problems. This is because duality principles relate a pair of optimization problems, known as primal and dual problems, in such a way that the existence of solution of one ensures the solution of the other and the extrema of two problems coincide under certain conditions. The duality theorems for linear optimization were conjectured by Neumann [1]. Thereafter, Wolfe [2] introduced a dual model known as “Wolfe dual” for a class of nonlinear convex optimization problems and established the duality results. Later, Mond and Weir [3] introduced the “Mond-Weir dual” whose objective function is simple as compared to the objective function of the Wolfe dual. After that, many researchers obtained duality results for various optimization problems under different assumptions. Mond and Smart [4] studied the control problem and derived the duality results under the invexity assumption. Thereafter, Mititelu [5] formulated Wolfe’s dual for the multi-time control problem and established the duality results under the invexity hypotheses. Beck and Ben-Tal [6] investigated the solution of the uncertain optimization problem via duality and shown that the primal worst case is equal to the dual best case. Jeyakumar et al. [7] also studied the robust duality for uncertain optimization problem under generalized convexity.