Synthesis, Analysis and Algorithms of Optimization and Equilibrium Systems

优化和平衡系统的综合、分析和算法

• 批准号: 11694151
• 负责人: FUKUSHIMA Masao
• 金额: $ 6.34万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11694151/
• 关键词: Optimization; Equilibrium model; Algorithm; Mathematical programming; Complementarity problem; Variation inequality problem; Parallel algorithm

Optimization is a fundamental concept in systems approach and a variety of methods have been developed and applied to practical problems. However, in order to deal with more complex problems that arise in engineering and social sciences, it is necessary to introduce the concept of hierarchical optimization or more generally optimization under equilibrium conditions, and construct a framework of effective methods for such problems. From this standpoint, the project have developed a number of approaches to the solution of various optimization and equilibrium problems. The obtained results are summarized as follows :1. Efficient algorithms for solving the system of nonlinear equations, which constitutes the most fundamental class of equilibrium. Moreover efficient algorithms for unconstrained and constrained optimization problems have been developed.2. The proximal point algorithm, which is one of the most basic approaches to optimization and equilibrium problems, has been studied, and some novel results on its desirable convergence properties have been established.3. Various reformulation methods that enable us to solve equilibrium problems by means of optimization techniques have been developed.4. Optimization problems involving equilibrium constraints have been studies in depth and a number of novel and remarkable results have been obtained.

优化是系统方法中的一个基本概念，并且已经开发了多种方法并将其应用于实际问题。但是，为了处理工程和社会科学中出现的更复杂的问题，有必要在平衡条件下介绍层次优化的概念或更普遍的优化，并为这些问题构建有效方法的框架。从这个角度来看，该项目开发了许多方法来解决各种优化和均衡问题。所获得的结果总结如下：1。用于解决非线性方程系统的有效算法，该算法构成了最基本的平衡类别。此外，已经开发了无约束和约束优化问题的有效算法。2。已经研究了近端点算法，它是优化和均衡问题的最基本方法之一，并且已经建立了有关其理想收敛性能的一些新结果。3。已经开发了各种使我们能够通过优化技术解决均衡问题的重新制定方法。4。涉及均衡约束的优化问题是深入研究，并且已经获得了许多新颖和显着的结果。

项目成果

J.P. Zeng, D. Li and M. Fukushima: "Weighted max-norm estimate of additive Schwarz iteration scheme for solving linear complementarity problems"Journal of Computational and Applied Mathematics. Vol.131,No.1. 1-14 (2001)

J.P. Zeng、D. Li 和 M. Fukushima：“用于解决线性互补问题的加法 Schwarz 迭代方案的加权最大范数估计”计算与应用数学杂志。

D.Li and M.Fukushima: "A globally and suporlinearly convergent Gauss-Newton based BFGS method for symmetric equations"SIAM Journal on Numerical Analysis. (掲載予定).

D.Li 和 M.Fukushima：“基于对称方程的全局超线性收敛高斯-牛顿 BFGS 方法”SIAM 数值分析杂志（即将出版）。

M. Fukushiina and P. Tseng: "An implementable active-set algorithm for computing a B-stationary point of the mathematical program with linear complementarity constraints"SIAM Journal on Optimization. Vol.12,No.3. 724-739 (2001)

M. Fukushiina 和 P. Tseng：“一种可实现的活动集算法，用于计算具有线性互补约束的数学程序的 B 驻点”SIAM 优化杂志。

D.Li, N.Yamashita, M.Fukushima: "A nonsmooth equation based BFGS method for solving KKT systems in mathematical programming"Journal of Optimization Theory and Applications. 109・1. 123-167 (2001)

D.Li、N.Yamashita、M.Fukushima：“数学规划中求解 KKT 系统的基于非光滑方程的 BFGS 方法”优化理论与应用杂志 109・1（2001 年）。

J.S.Pang and M.Fukushima: "Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints"Computational Optimization and Applications. 13. 111-136 (1999)

J.S.Pang 和 M.Fukushima：“具有平衡约束的数学程序的互补约束资格和简化 B 平稳条件”计算优化和应用。