Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems

Due to their fundamental nature and numerous applications, sphere constrained polynomial optimization problems have received a lot of attention lately. In this paper, we consider three such problems: (i) maximizing a homogeneous polynomial over the sphere; (ii) maximizing a multilinear form over a Cartesian product of spheres; and (iii) maximizing a multiquadratic form over a Cartesian product of spheres. Since these problems are generally intractable, our focus is on designing polynomial–time approximation algorithms for them. By reducing the above problems to that of determining the L2–diameters of certain convex bodies, we show that they can all be approximated to within a factor of Ω((logn/n)d/2−1) deterministically, where n is the number of variables and d is the degree of the polynomial. This improves upon the currently best known approximation bound of Ω((1/n)d/2−1) in the literature. We believe that our approach will find further applications in the design of approximation algorithms for polynomial optimization problems with provable guarantees.