A dynamic inequality generation scheme for polynomial programming

Abstract

Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In this paper, we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs. When used iteratively, this scheme improves the bounds without incurring an exponential growth in the size of the relaxation. As a result, the proposed scheme is in principle scalable to large general polynomial programming problems. When all the variables of the problem are non-negative or when all the variables are binary, the general algorithm is specialized to a more efficient algorithm. In the case of binary polynomial programs, we show special cases for which the proposed scheme converges to the global optimal solution. We also present several examples illustrating the computational behavior of the scheme and provide comparisons with Lasserre’s approach and, for the binary linear case, with the lift-and-project method of Balas, Ceria, and Cornuéjols.