On the geometric analysis of a quartic-quadratic optimization problem under a spherical constraint

• 成果类型: Article
• 署名作者: Zhang, Haixiang; Milzarek, Andre; Wen, Zaiwen; Yin, Wotao
• 署名单位: University of California System; University of California Berkeley; The Chinese University of Hong Kong, Shenzhen; Shenzhen Research Institute of Big Data; Shenzhen Institute of Artificial Intelligence & Robotics for Society; Peking University; Peking University; University of California System; University of California Los Angeles
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01702-6
• 发表日期: 2022
• 页码: 421-473
• 关键词: optimality conditions; programming-problems; proximal algorithm; numerical-solution; matrix completion; descent methods; error-bounds; CONVERGENCE; nonconvex; minimization

摘要

This paper considers the problem of solving a special quartic-quadratic optimization problem with a single sphere constraint, namely, finding a global and local minimizer of 1/2z*Az + beta/2 Sigma(n)(k=1) vertical bar z(k)vertical bar(4) such that parallel to z parallel to(2) = 1. This problem spans multiple domains including quantum mechanics and chemistry sciences and we investigate the geometric properties of this optimization problem. Fourth-order optimality conditions are derived for characterizing local and global minima. When the matrix in the quadratic term is diagonal, the problem has no spurious local minima and global solutions can be represented explicitly and calculated in 0(n log n) operations. When A is a rank one matrix, the global minima of the problem are unique under certain phase shift schemes. The strict-saddle property, which can imply polynomial time convergence of secondorder-type algorithms, is established when the coefficient of the quartic term is either at least O(n(3/2)) or not larger than O(1). Finally, the Kurdyka-Lojasiewicz exponent of quartic-quadratic problem is estimated and it is shown that the largest exponent is at least 1/4 for a broad class of stationary points.