On cones of nonnegative quartic forms

Bo Jiang, Zhening Li, Shuzhong Zhang

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Abstract

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 higherdegree 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.
Original languageEnglish
Pages (from-to)161-197
JournalFoundations of Computational Mathematics
Volume17
Issue number1
Early online date30 Nov 2015
DOIs
Publication statusPublished - Feb 2017

Keywords

  • cone of polynomial functions
  • super-symmetric tensors
  • nonnegative quartic forms
  • sums of squares
  • SOS-convexity
  • polynomial optimization

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