A Bounded Degree Lasserre Hierarchy with SOCP Relaxations for Global Polynomial Optimization and Applications

Abstract

In this paper, we propose a new convergent conic programming hierarchy of relaxations involving both semi-definite cone and second-order cone constraints for solving nonconvex polynomial optimization problems to global optimality. The significance of this hierarchy is that the size and number of the semi-definite and second-order cone constraints of the relaxations are fixed and independent of the step or level of the approximation in the hierarchy. Using the Krivine-Stengle's certificate of positivity in real algebraic geometry, we establish the convergence of the hierarchy of relaxations, extending the very recent so-called bounded degree Lasserre hierarchy. In particular, we also provide a convergent bounded degree second-order cone programming (SOCP) hierarchy for solving polynomial optimization problems. We then present finite convergence at step one of the SOCP hierarchy for two classes of polynomial optimization problems: a subclass of convex polynomial optimization problems where the objective and constraint functions are SOCP-convex polynomials, defined in terms of specially structured sum of squares polynomials, and a class of polynomial optimization problems, involving polynomials with essentially non-positive coefficients. In the case of one-step convergence for problems with SOCP-convex polynomials, we show how a global solution is recovered from the relaxation via Jensen's inequality of SOCP-convex polynomials. As an application, we derive a corresponding convergent conic linear programming hierarchy for conic-convex semi-algebraic programs. Whenever the semi-algebraic set of the conic-convex program is described by convex polynomial inequalities, we show further that the values of the relaxation problems converge to the common value of the convex program and its Lagrangian dual under a constraint qualification.