求解约束优化问题的两个微分方程算法  

Two Differential Equation Algorithms for Solving Constrained Optimization Problems

作　　者：金丽[1,2] 肖现涛[1] 张立卫[1] 

机构地区：[1]大连理工大学应用数学系,大连116024 [2]浙江海洋学院数理与信息学院

出　　处：《运筹学学报》2007年第2期73-82,共10页Operations Research Transactions

基　　金：国家自然科学基金(10471015);归国留学人员科研启动基金.

摘　　要：本文给出求解具有等式约束和不等式约束的非线性优化问题的一阶信息和二阶信息的两个微分方程系统,问题的局部最优解是这两个微分方程系统的渐近稳定的平衡点,给出了这两个微分方程系统的Euler离散迭代格式并证明了它们的收敛性定理,用龙格库塔法分别求解两个微分方程系统．我们构造了搜索方向由两个微分系统计算,步长采用Armijo线搜索的算法分别求解这个约束最优化问题,在局部Lipschitz条件下基于二阶信息的微分方程系统的迭代方法具有二阶的收敛速度。我们给出的数值结果表明龙格库塔的微分方程算法具有较好的稳定性和更高的精确度,求解二阶信息的微分方程系统的方法具有更快的收敛速度．This paper presents two differential systems, involving first and the sec- ond order information on problem functions, respectively, for solving constrained nonlinear optimization problems with both equality and inequality constraints＇. The local minimum points of the problems are their asymptotically stable equilibrium points. The Euler discrete schemes for the both diiterential systems are presented and their convergence theorems are demonstrated, and the Runge-Kutta method is employed to solve these two differential equation systems. We construct algorithms in which directions are computed by these two systems and the step-sizes are generated by Armijo line search to solve the original constrained optimization problem and prove that the discrete scheme based on the differential equation system with the second order information has the locally quadratic convergence rate under the local Lipschitz condition. The numerical results given here show that Runge-Kutta method has better stability and higher precision and the numerical method based on the differential equation system with the second information is faster than the other one.

关 键 词：运筹学 约束非线性优化 约束规范 微分方程 渐近稳定性 平衡点 

分 类 号：O175.14[理学—数学]