Approximation Methods for Inhomogeneous Polynomial Optimization

In this paper, we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a compact set. The focus is on the design and analysis of polynomial-time approximation algorithms. The references on approximation algorithms for inhomogeneous polynomial optimization problems are extremely scarce in the literature. To the best of our knowledge, the only result so far was due to Nemirovski, Roos and Terlaky [26], who obtained an Ω (1/ logm)-approximation ratio for maximizing an inhomogeneous quadratic polynomial over the intersection of m co-centered ellipsoids. In this paper we aim at developing computational methods to deal with optimization models with polynomial objective functions in any fixed degrees. In particular, we first study the problem of maximizing an inhomogeneous polynomial over the Euclidean ball. A polynomial-time approximation algorithm is proposed for this problem with an assured (relative) worst-case performance ratio, which is dependent only on the dimensions of the model. The method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co-centered ellipsoids. Finally, the constraint set is extended to a general compact set. Specifically, we propose a polynomial-time approximation algorithm with a (relative) worst-case performance ratio for polynomial optimization over some convex compact sets, e.g., a polytope.