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In this paper, we consider robust optimal solutions for a convex optimization problem in the face of data uncertainty both in the objective and constraints. By using the properties of the subdifferential sum formulae, we first introduce a robust-type subdifferential constraint qualification, and then obtain some completely characterizations of the robust optimal solution of this uncertain convex optimization problem. We also investigate Wolfe type robust duality between the uncertain convex optimization problem and its uncertain dual problem by proving duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. Moreover, we show that our results encompass as special cases some optimization problems considered in the recent literature.

In this paper, by using the properties of the epigraph of the conjugate functions, we introduce some closedness conditions and investigate some characterizations of these closedness conditions. Then, by using these closedness conditions, we obtain some Farkas-type results for a constrained fractional programming problem with DC functions. We also show that our results encompass as special cases some programming problems considered in the recent literature.

In this paper, we deal with extended Ky Fan inequalities (EKFI) with DC functions. Firstly, a dual scheme for (EKFI) is introduced by using the method of Fenchel conjugate function. Under suitable conditions, weak and strong duality assertions are obtained. Then, by using the obtained duality assertions, some Farkas-type results which characterize the optimal value of (EKFI) are given. Finally, as applications, the proposed approach is applied to a convex optimization problem (COP) and a generalized variational inequality problem (GVIP).

In this paper, by virtue of lower Studniarski derivatives of set-valued maps, relationships between lower Studniarski derivative of a set-valued map and its profile map are discussed. Some results concerning sensitivity analysis are obtained in parametrized vector optimization.