多项式优化简介  

An introduction to polynomial optimization

作　　者：黄磊 聂家旺 袁亚湘[2] Lei Huang;Jiawang Nie;Ya-Xiang Yuan

机构地区：[1]Department of Mathematics,University of California San Diego,La Jolla,CA 92093,USA [2]中国科学院数学与系统科学研究院,北京100190

出　　处：《中国科学:数学》2025年第2期481-500,共20页Scientia Sinica:Mathematica

摘　　要：多项式优化是目标函数和约束函数均为多项式的一类特殊的非线性优化问题.矩-平方和松弛算法是全局求解多项式优化问题的主要算法.本文主要介绍多项式优化的矩-平方和松弛以及其有限收敛性理论、全局最优性的判别以及如何获取全局最优解.当标准的矩-平方和松弛不具有收敛性时,本文介绍如何利用Lagrange乘子表示来构造具有有限收敛性的矩-平方和松弛.Polynomial optimization is a broad class of optimization problems whose objective functions and constraining functions are polynomials.The moment-SOS(sum of squares)hierarchy is the most powerful method to solve polynomial optimization globally.In this paper,we introduce moment-SOS methods for solving polynomial optimization.We also study their finite convergence theory,the certificate for global optimality,and how to extract minimizers.When the standard moment-SOS hierarchy does not have finite convergence,we introduce tight relaxations by using Lagrange multiplier expressions.

关 键 词：多项式优化 矩-平方和松弛 有限收敛 最优性条件 Lagrange乘子表示 

分 类 号：O224[理学—运筹学与控制论]