Sos Approximations of Nonnegative Polynomials via Simple High Degree Perturbations

Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate the analysis of optimization problems over the boolean hypercube via a recent, ...

A Complete Characterization of the Gap between Convexity and SOS-Convexity

Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials via the definition of convexity, its first order characterization, and its second order characterization are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming w...

High Degree Perturbations of Nonnegative Polynomials

A convex polynomial that is not sos-convex

A multivariate polynomial p(x) = p(x1, . . . , xn) is sos-convex if its Hessian H(x) can befactored as H(x) = M (x)M(x) with a possibly nonsquare polynomial matrix M(x). It iseasy to see that sos-convexity is a sufficient condition for convexity of p(x). Moreover, theproblem of deciding sos-convexity of a polynomial can be cast as the feasibility of a semidefiniteprogram, wh...

Refined Asymptotics for Sparse Sums of Squares

To prove that a polynomial is nonnegative on R one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods (see, e.g., [Par03]), thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact...