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 many of them are found easy to deal with. Limitations of the quadratics, however, become increasingly binding as higher-degree nonlinearity is imperative in modern applications of optimization. In recent years, one observes a surge of research activities in polynomial optimization, and modeling with quartic or higher-degree 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. (Math Program Ser A 137:453–476, 2013) proved that checking the convexity of a quartic polynomial is strongly NP-hard in general, which settles a long-standing open question. In this paper, we proceed to study six fundamentally important convex cones of quartic forms in the space of super-symmetric tensors, including the cone of nonnegative quartic forms, the sums of squared forms, the convex quartic forms, and the sums of fourth-power forms. It turns out that these convex cones coagulate into a chain in a decreasing order with varying complexity status. Potential applications of these results to solve highly nonlinear and/or combinatorial optimization problems are discussed.
Bibliographical noteFunding Information:
We would like to thank three anonymous referees for their insightful comments, which helped significantly improve this paper from its original version. This work was partially supported by National Science Foundation of China (Grants 11401364 and 11371242) and the US National Science Foundation (Grant CMMI-1161242).
© 2015, SFoCM.
- Cone of polynomial functions
- Nonnegative quartic forms
- Polynomial optimization
- Sums of squares
- Super-symmetric tensors