Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere
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We study the problem of maximizing the geometric mean of d low-degree non-negative forms on the real or complex sphere in n variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is equivalent to optimizing a homogeneous polynomial of degree O(d) on the sphere. The standard Sum-of-Squares based convex relaxation for this polynomial optimization problem requires solving a semidefinite program (SDP) of size n^O(d), with multiplicative approximation guarantees of Ω(1/n). We exploit the compact representation of this polynomial to introduce a SDP relaxation of size polynomial in n and d, and prove that it achieves a constant factor multiplicative approximation when maximizing the geometric mean of non-negative quadratic forms. We also show that this analysis is asymptotically tight, with a sequence of instances where the gap between the relaxation and true optimum approaches this constant factor as d →∞. Next we propose a series of intermediate relaxations of increasing complexity that interpolate to the full Sum-of-Squares relaxation, as well as a rounding algorithm that finds an approximate solution from the solution of any intermediate relaxation. Finally we show that this approach can be generalized for relaxations of products of non-negative forms of any degree.
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