Homogeneous polynomials and spurious local minima on the unit sphere

被引:0
|
作者
Jean B. Lasserre
机构
[1] University of Toulouse,LAAS
来源
Optimization Letters | 2022年 / 16卷
关键词
Homogeneous polynomials; Global and local minima; Optimization on the unit sphere;
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摘要
We consider forms on the Euclidean unit sphere. We obtain a simple and complete characterization of all points that satisfies the standard second-order necessary condition of optimality. It is stated solely in terms of the value of (i) f, (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. We also characterize a class of degree-d forms with no spurious local minima on Sn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{n-1}$$\end{document} by using a property of gradient ideals in algebraic geometry.
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页码:1105 / 1118
页数:13
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