On Cones of Nonnegative Quartic Forms

Historically, much of the theory and practice in nonlinear optimization has revolved around the quadratic models. Though quadratic functions are nonlinear polynomials, they are well structured and easy to deal with. Limitations of the quadratics, however, become increasingly binding as higher degree nonlinearity is imperative in modern applications of optimization. In the recent years, one observes a surge of research activities in polynomial optimization, and modeling with quartic or higher order polynomial functions has been more commonly accepted. On the theoretical side, there are also major recent progresses on polynomial functions and optimization. For instance, Ahmadi et al. [2] proved that checking the convexity of a quartic polynomial function is strongly NP-hard in general, which settles a long-standing open question. In this paper we proceed to studying six fundamentally important convex cones of quartic functions in the space of symmetric quartic tensors, including the cone of nonnegative quartic polynomials, the sum of squared polynomials, the convex quartic polynomials, and the sum of fourth powered polynomials. It turns out that these convex cones coagulate into a chain in decreasing order. The complexity status of these cones is sorted out as well. Finally, potential applications of the new results to solve highly nonlinear and/or combinatorial optimization problems are discussed.