Certifying the Global Optimality of Quartic Minimization over the Sphere  

作　　者：Sheng-Long Hu 

机构地区：[1]Department of Mathematics,School of Science,Hangzhou Dianzi University,Hangzhou 310018,Zhejiang,China

出　　处：《Journal of the Operations Research Society of China》2022年第2期241-287,共47页中国运筹学会会刊（英文）

基　　金：This work is partially supported by the National Natural Science Foundation of China(No.11771328);Young Elite Scientists Sponsorship Program by Tianjin,and the Natural Science Foundation of Zhejiang Province,China(No.LD19A010002).

摘　　要：The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global optimal solution was found but without a certification.We will present in this article two classes of methods which are able to certify the global optimality,i.e.,algebraic methods and semidefinite program(SDP)relaxation methods.Several advances on these topics are summarized,accompanied with some emerged new results.We want to emphasize that for mediumor large-scaled instances,the problem is still a challenging one,due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.

关 键 词：Quartic minimization Tensor SPHERE Global optimality Elimination method Critical points EIGENVECTORS Determinant NONDEGENERATE Characteristic polynomial SDP relaxations Moment matrix Flatness POSITIVSTELLENSATZ Nonnegative polynomial Sums of squares Duality 

分 类 号：O22[理学—运筹学与控制论]