Studies on Development and Synthesis of Efficient Algorithms for Optimization and Equilibrium Problems

优化和平衡问题的高效算法的开发和综合研究

• 批准号: 08650079
• 负责人: FUKUSHIMA Masao
• 金额: $ 1.41万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08650079/
• 关键词: Optimization; Equilibrium Model; Algorithm; Variational Inequality Problem; Complementarity Problem

Variational inequality and complementarity problems are fundamental problems that are very useful in formulating various equilibrium problems arising not only in engineering such as transportation, structural mechanics and operations reseach but also in social sciences such as economics. In this project, we have conducted research on those fundamental equilibrium problems with particular emphasis on the optimization reformulation approach that have recently attracted much attention in this field. The main results which have been obtained during the last two years are summarized as follows :1.New approaches that reformulate variational inequality and complementarity problems as equivalent optimization problems have been proposed. Moreover novel algorithms that extend descent methods and Newton's method have been developed, and their effectiveness has been examined through extensive numerical experiments.2.New parallel and decomposition methods for variational inequality and complementarity problems have been developed by meking use of the idea of operator splitting and matrix splitting.3.New algorithms have been developed for solving mathematical program with equilibrium constraints for which a unified study has not been done so far.

变分不等式和互补性问题是非常有用的基本问题，对于表述各种均衡问题非常有用，这些均衡问题不仅出现在交通运输、结构力学和运筹学等工程领域，而且还出现在经济学等社会科学领域。在这个项目中，我们对那些基本均衡问题进行了研究，特别强调了最近在该领域引起广泛关注的优化重构方法。近两年来取得的主要成果概括如下： 1.提出了将变分不等式和互补问题重新表述为等价优化问题的新方法。此外，还开发了扩展下降法和牛顿法的新算法，并通过大量数值实验检验了它们的有效性。2.利用算子分裂的思想，开发了变分不等式和互补问题的新并行和分解方法3.已经开发了新的算法来求解具有平衡约束的数学规划，但迄今为止尚未进行统一的研究。

项目成果

M.Fukushima: "Minimizing and stationary sequences of merit functions for complementarity problems and variational inequalities" Complementarity and Variational Problems:State of the Art. (発表予定).

M.Fukushima：“互补性问题和变分不等式的评价函数的最小化和平稳序列”互补性和变分问题：最新技术（待提交）。

N.Yamashita and M.Fukushima: "Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems" Mathematical Programming. 76. 469-491 (1997)

N.Yamashita 和 M.Fukushima：“用于解决单调互补问题的半平滑重构的改进牛顿法”数学规划。

H.Jiang, M.Fukushima, L.Qi and D.Sun: "A trust region method for solving generalized complementarity problems" SIAM Journal on Optimization. (掲載予定).

H.Jiang、M.Fukushima、L.Qi 和 D.Sun：“解决广义互补问题的信任域方法”SIAM Journal on Optimization（即将出版）。

C.Kanzow: "Equivalence of the generalized complementarity problem to differentiable unconstrained minimization" J.Opt.Theory and Appl.90・3. 581-603 (1996)

C. Kanzow：“广义互补问题与可微无约束最小化的等价”J.Opt.Theory and Appl.90・3 (1996)。

N.Yamashita and M.Fukushima: "Equivalent unconstrained minimization and global error bounds for variational inequality problems" SIAM Journal on Control and Optimization. 35. 273-284 (1977)

N.Yamashita 和 M.Fukushima：“变分不等式问题的等效无约束最小化和全局误差界限”SIAM 控制与优化杂志。