Characterizing a Class of Robust Vector Polynomial Optimization via Sum of Squares Conditions

This paper deals with an SOS-convex (sum of squares convex) polynomial optimization problem with spectrahedral uncertain data in both the objective and constraints. By using a robust-type characteristic cone constraint qualification, we first obtain necessary and sufficient conditions for robust weakly efficient solutions of this uncertain SOS-convex polynomial optimization problem in terms of sum of squares conditions and linear matrix inequalities. Then, we propose a relaxation dual problem for this uncertain SOS-convex polynomial optimization problem and explore weak and strong duality properties between them. Moreover, we give a numerical example to show that the relaxation dual problem can be reformulated as a semidefinite linear programming problem.