Journal Title
Title of Journal: Found Comput Math
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Abbravation: Foundations of Computational Mathematics
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Authors: Bo Jiang Zhening Li Shuzhong Zhang
Publish Date: 2015/11/30
Volume: 17, Issue: 1, Pages: 161-197
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 higherdegree 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 137453–476 2013 proved that checking the convexity of a quartic polynomial is strongly NPhard in general which settles a longstanding open question In this paper we proceed to study six fundamentally important convex cones of quartic forms in the space of supersymmetric tensors including the cone of nonnegative quartic forms the sums of squared forms the convex quartic forms and the sums of fourthpower 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 discussedWe 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 CMMI1161242To start with let us first investigate the feasible regions of these two problems to be denoted by mathrmfeasQ and mathrmfeasRQ respectively The relationship between mathrmfeasQ and mathrmfeasRQ is revealed by the following lemma
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