Weighted robust optimality of convex optimization problems with data uncertainty

In this paper, we introduce the weighted robust counterpart of convex optimization problem with data uncertainty both in the objective and constraints. Then, optimal solutions of the weighted robust optimization problem are weakly Pareto optimal solutions of an unconstrained multicriteria optimization problem, and these solutions are Pareto optimal solutions under the uniqueness assumption. We also prove that the intersection of the optimal solution set of the weighted robust optimization problem and the Pareto optimal solution set of the unconstrained multicriteria optimization problem is nonempty. Finally, optimality conditions of the weighted robust optimal solution for the uncertain convex optimization problem are also obtained under some suitable assumptions.