The AHI family of sum of squares polynomials

Abstract

We introduce a new family of non-negative polynomials, constructed via the arithmetic harmonic inequality, called AHI polynomials. We derive explicit algebraic conditions for this family and prove that, for AHI polynomials, the cone of non-negative polynomials coincides with the cone of sum of squares (SOS) polynomials. We then study their convexity, showing that although AHI polynomials are generally nonconvex, certain monomial substructures are SOS convex. We further locate the family precisely among the standard nonnegativities certificates; every AHI polynomial is simultaneously SOS and a sum of non negative circuit polynomials (SONC), and the containment in the intersection of these two cones is strict. By closing this family under multiplication, we obtain a cone Pi AHI that is, by construction, still SOS, yet we prove that it lies outside both the SONC cone and the smaller SDSOS cone. Moreover, membership in this cone admits a closed form certificate that does not require solving any semidefinite programs. Finally, we demonstrate the usefulness of these structures in optimization, numerical experiments indicate that exploiting AHI sparsity yields a computation time over 300 times faster than dense SOS relaxations and enables solving high degree polynomial optimization problems (up to degree 40) that standard methods cannot handle due to computational limits, and a factorized hierarchy for Pi AHI decomposes products into independent small subproblems that generic sparsity techniques do not detect.