Linear Matrix Inequality Conditions and Duality for a Class of Robust Multiobjective Convex Polynomial Programs

This paper is concerned with an uncertain multiobjective optimization problem, where the objective and constraint functions are sums-of-squares--convex (SOS-convex) polynomials, and the uncertainty sets are spectrahedra. Using a robust optimization approach, we first establish necessary and sufficient optimality conditions for weakly efficiencies of the corresponding robust multiobjective optimization problem. These optimality criteria are expressed in terms of sums-of-squares conditions and linear matrix inequalities, which provide a numerically checkable certificate of the solvability of the given optimality conditions. We then propose a dual multiobjective problem by means of sums-of-squares and linear matrix inequalities to the robust multiobjective SOS-convex polynomial optimization problem and examine weak, strong, and converse duality relations between them. In addition, we consider a semidefinite linear programming (SDP) weighted-sum relaxation problem for verifying weighted-sum efficient values of the primal problem.