Optimality conditions for quasi ($epsilon,alpha$)-solutions in convex optimization problems under data uncertainty (Study on Nonlinear Analysis and Convex Analysis)

Abstract

Under the robust characteristic cone constraint qualification (RCCCQ), a represented form of the epsilon-normal set to a convex set C at overline{x}in C is proposed, where C : ={xin mathbb{R}^{n}:g(cdot, v)leqq 0, forall vin mathcal{V}}, here g : mathbb{R}^{n}cross mathbb{R}^{p}arrow mathbb{R} is a continuous function such that for all vin mathbb{R}^{p}, g(cdot, v) is a convex function, and mathcal{V}subset mathbb{R}^{p} is a compact and convex uncertain set. Then, the proposed result is applied to studying approximate optimality theorems for a quasi (alpha, epsilon)-solution for the robust counterpart of a convex optimization problem in the face of data uncertainty.