On approximate solutions and saddle point theorems for robust convex optimization

This paper provides some new results on robust approximate optimal solutions for convex optimization problems with data uncertainty. By using robust optimization approach (worst-case approach), we first establish necessary and sufficient optimality conditions for robust approximate optimal solutions of this uncertain convex optimization problem. Then, we introduce a Wolfe-type robust approximate dual problem and investigate robust approximate duality relations between them. Moreover, we obtain some robust approximate saddle point theorems for this uncertain convex optimization problem. We also show that our results encompass as special cases some optimization problems considered in the recent literature.