LAGRANGIAN-CONIC RELAXATIONS, PART II: APPLICATIONS TO POLYNOMIAL OPTIMIZATION PROBLEMS

We present a moment cone (MC) relaxation and a hierarchy of Lagrangian-SDP relaxations of polynomial optimization problems (POPs) using the unified framework established in Part I. The MC relaxation is derived for a POP of minimizing a polynomial subject to a nonconvex cone constraint and polynomial equality constraints. It is an extension of the completely positive programming relaxation for QOPs. Under a copositivity condition, we characterize the equivalence of the optimal values between the POP and its MC relaxation. A hierarchy of Lagrangian-SDP relaxations, which is parameterized by a positive integer w, is proposed for an equality constrained POP. It is obtained by combining Lasserre's hierarchy of SDP relaxation of POPs and the basic idea behind the conic and Lagrangian-conic relaxations from the unified framework. We prove under a certain assumption that the optimal value of the Lagrangian-SDP relaxation with the Lagrangian multiplier lambda and the relaxation order w in the hierarchy converges to that of the POP as lambda ->infinity and omega ->infinity The hierarchy of Lagrangian-SDP relaxations is designed to be used in combination with the bisection and 1-dimensional Newton methods, which was proposed in Part I, for solving large-scale POPs efficiently and effectively.