约束优化的内点增广Lagrange函数方法  

Interior-point augmented Lagrangian methods for constrained optimization

作　　者：刘新为 戴彧虹[2] Xinwei Liu;Yu-Hong Dai

机构地区：[1]河北工业大学数学研究院,天津300401 [2]中国科学院数学与系统科学研究院计算数学与科学工程计算研究所,北京100190

出　　处：《中国科学:数学》2025年第2期535-550,共16页Scientia Sinica:Mathematica

基　　金：中国科学院战略性先导科技专项(A类)(批准号:XDA27010101);国家自然科学基金(批准号:12471286,12071108,12021001和11991021)资助项目。

摘　　要：内点方法是求解不等式约束优化的一类重要和有效的方法,而经典增广Lagrange函数方法是求解等式约束优化和凸优化的有效方法.最近的研究表明,结合内点技术的增广Lagrange函数具有很好的高阶光滑性质,并能够保持优化问题的凸性.因此该函数既能够用于有效求解非凸优化,也能够用于有效求解凸优化问题.本文综述近年来我们关于内点增广Lagrange函数方法的系列研究进展,并对下一步的研究方向进行展望.The interior-point approach is one kind of the most important and effective methods for nonlinear inequality-constrained optimization,whereas classic augmented Lagrangian methods are very efficient in solving nonlinear equality-constrained optimization and convex optimization.The new research has shown that the interior-point augmented Lagrangian function can be of very nice higher-order smoothness,and is convex when the original optimization problem is convex.These nice properties make the novel augmented Lagrangian applicable and effective not only for convex optimization but also for non-convex optimization problems.In this paper,we provide an overview of our recent research progress about the interior-point augmented Lagrange function methods on the non-convex and convex optimization,and point out prospects for future research directions.

关 键 词：约束优化 非凸优化 凸优化 线性规划 增广LAGRANGE函数 内点方法 交替极小化方法 

分 类 号：O224[理学—运筹学与控制论]