Finding Efficient Solutions for Multicriteria Optimization Problems with SOS-convex Polynomials

Abstract

In this paper, we focus on the study of finding efficient solutions for a multicriteria optimization problem (MP), where both the objective and constraint functions are SOS-convex polynomials. By using the well-known $\epsilon$-constraint method (a scalarization technique), we substitute the problem (MP) to a class of scalar ones. Then, a zero duality gap result for each scalar problem, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programming problem, is established, under a suitable regularity condition. Moreover, we prove that an optimal solution of each scalar problem can be found by solving its associated semidefinite programming problem. As a consequence, we show that finding efficient solutions for the problem (MP) is tractable by employing the $\epsilon$-constraint method. A numerical example is also given to illustrate our results.