Convergent hierarchy of SDP relaxations for a class of semi-infinite convex polynomial programs and applications

In this paper we examine semidefinite linear programming approximations to a class of semi-infinite convex polynomial optimization problems, where the index sets are described in terms of convex quadratic inequalities. We present a convergent hierarchy of semidefinite linear programming relaxations under a mild well-posedness assumption. We also provide additional conditions under which the hierarchy exhibits finite convergence. These results are derived by first establishing characterizations of optimality which can equivalently be reformulated as linear matrix inequalities. A separation theorem of convex analysis and a sum-of-squares polynomial representation of positivity of real algebraic geometry together with a special variable transformation play key roles in achieving the results. Finally, as applications, we present convergent relaxations for a broad class of robust convex polynomial optimization problems.