微分方程方法求解约束优化问题
Differential Equation Method for Solving Constrained Optimization Problems
摘要: 本文探讨了微分方程方法在求解约束优化问题的应用，讨论解的收敛性和收敛速度。首先，通过对原始约束优化所对应的Karush-Kuhn-Tucker条件进行转换后，利用光滑互补函数，将问题转化成求解光滑方程组S(ε,x,μ,λ)=0，进一步转化成无约束优化问题。我们利用了微分方程系统来求解最终的无约束优化问题，并在一定的约束条件下，得到了该微分方程系统的解稳定性及收敛速度，从而得到了所求约束优化问题的收敛性和解的收敛速度。最后，给出数值实验说明所提出的微分方程方法求解约束优化问题的有效性。
Abstract: This article explores the application of differential equation methods in solving constrained optimization problems, and discusses the convergence and speed of solutions. Firstly, by transforming the Karush Kuhn Tucker condition corresponding to the original constrained optimization, and using a smooth complementarity function, the problem is transformed into solving a smooth equation systemS(ε,x,μ,λ)=0, which is further transformed into an unconstrained optimization problem. We utilized a differential equation system to solve the final unconstrained optimization problem, and under certain constraint conditions, obtained the solution stability and convergence rate of the differential equation system, thus obtaining the convergence and convergence rate of the constrained optimization problem. Finally, numerical experiments are provided to demonstrate the effectiveness of the proposed differential equation method for solving constrained optimization problems.
文章引用：谢红俭, 孙菊贺, 王莉, 吕琪楠. 微分方程方法求解约束优化问题[J]. 应用数学进展, 2024, 13(5): 2125-2132. https://doi.org/10.12677/aam.2024.135200

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