约束优化问题的一种基于双目标策略的增广Lagrange算法  

Augmented Lagrange Algorithm Based Bi-object Strategy for Constraint Optimization Problem

作　　者：王艺文 贺素香[1] WANG YIWEN;HE SUXIANG(School of Science,Wuhan University of Technology,Wuhan 430070,China)

机构地区：[1]武汉理工大学数学系,武汉430070

出　　处：《应用数学学报》2021年第6期763-779,共17页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(11671183);武汉理工大学自主创新项目(2019IB010,2020IB010)资助项目.

摘　　要：针对传统增广Lagrange方法中精确求解子问题计算量大的问题,基于Rockafellar提出的求解不等式约束优化问题的增广Lagrange函数,本文提出了一种具有双目标策略的增广Lagrange算法.每次迭代时,首先极小化增广Lagrange函数的二次近似函数以得到搜索方向,进一步借助一个辅助信赖域子问题来判断这个搜索方向能否被接受.其次分别基于目标函数和约束违反度函数提出双目标策略以判断当前线搜索是否成功,其中一个策略是为了降低目标函数值,另一个策略是为了减少约束违反度.在一些假设条件下,分析了算法的可行性,并且在相对较弱的假设条件下,证明了算法的全局收敛性.最后,对经典算例进行数值实验并分析其实验结果.For the costs of obtaining the exact solutions in classical augmented Lagrangemethod being prohibitive from the computational point of view,this paper presents anaugmented Lagrange algorithm with bi-object strategy based on the augmented Lagrangefunction proposed by Rockafellar for solving inequality constrained optimization problems.At every iteration,the search direction is firstly obtained by minimizing a quadratic approx-imation function to the augmented Lagrange function and an auxiliary trust region subprob-lem is introduced to decide whether the search direction will be accepted or not.Secondly,a bi-object strategy is proposed to judge whether the current line search is successful ornot based on the objective function and the measure of constraint violation respectively,in which one is to reduce the value of objective function and the other is to decrease themeasure of constraint violation.Under some assumptions,the feasibility of the algorithmis analyzed and under the relatively weak assumptions,the global convergence of the algo-rithm is proved.Finally,the experimental results of the classical examples are analyzed bynumerical experiments.

关 键 词：约束优化问题 增广LAGRANGE函数 双目标策略 全局收敛性 

分 类 号：O221[理学—运筹学与控制论]