Multi-objective Optimization Problems with SOS-convex Polynomials over an LMI Constraint

In this paper, we aim to find efficient solutions of a multi-objective optimization problem over a linear matrix inequality (LMI in short), in which the objective functions are SOS-convex polynomials. We do this by using two scalarization approaches, that is, the 6-constraint method and the hybrid method. More precisely, we first transform the considered multi-objective optimization problem into their scalar forms by the 6-constraint method and the hybrid method, respectively. Then, strong duality results, between each formulated scalar problem and its associated semidefinite programming dual problem, are given, respectively. Moreover, for each proposed scalar problem, we show that its optimal solution can be found by solving an associated single semidefinite programming problem, under a suitable regularity condition. As a consequence, we prove that finding efficient solutions to the considered problem can be done by employing any of the two scalarization approaches. Besides, we illustrate our methods through some nontrivial numerical examples.