Studies on Convex Optimization and Related Problems

凸优化及相关问题的研究

基本信息

批准号：
14350046
负责人：
FUKUSHIMA Masao
金额：
$ 6.14万
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：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14350046/
关键词：
convex optimization mathematical programming algorithm complementarity problem iterative method 相補性問題

项目摘要

Convex optimization is a basic research area which has been studied for a long time. It is also a very hot research area in which various applications of practical importance have recently been discovered as novel efficient solution methods such as interior point algorithms have been developed. The aim of this research is to develop practically efficient methods based on solid theoretical ground for solving convex optimization problems and related problems involving in particular constrained convex programming problems, positive semi-definite programming problems, and monotone complementarity problems, thereby contributing to expand the area of applications in engineering. The methods developed in this project are listed as follows : (1)Sequential quadratically constrained quadratic programming method for constrained convex programming problems ; (2)Regularized Newton method for minimizing convex functions with possibly singular Hessians ; (3)Active set identification technique in the proximal point method for monotone complementarity problems ; (4)Iterative methods for mathematical programs with equilibrium constraints ; (5)Smoothing Newton method for second-order cone complementarity problems ; (6)Matrix splitting method for second-order cone complementarity problems. Moreover, we introduced a new equilibrium concept in a non-cooperative game and studied it through a second-order cone complementarity problem. We have also studied nonlinear semidefinite programming problems and mathematical programs with equilibrium constraints under uncertainty. These studies will lead to our next research subject of robust optimization.

凸优化是一个已经研究了很长时间的基础研究领域。这也是一个非常热门的研究领域，随着诸如内点算法之类的新颖有效的解决方法的开发，最近发现了各种具有实际重要性的应用。本研究的目的是基于坚实的理论基础开发实用有效的方法来解决凸优化问题和相关问题，特别是约束凸规划问题、正半定规划问题和单调互补问题，从而有助于扩大领域工程中的应用。本课题开发的方法如下： (1)约束凸规划问题的序贯二次约束二次规划方法； (2)正则牛顿法最小化可能具有奇异Hessians的凸函数； (3)单调互补问题的近点法中的活动集识别技术； (4)具有平衡约束的数学规划的迭代方法； (5)二阶锥互补问题的平滑牛顿法； (6)二阶锥互补问题的矩阵分裂法。此外，我们在非合作博弈中引入了新的均衡概念，并通过二阶锥体互补问题对其进行了研究。我们还研究了非线性半定规划问题和不确定性下具有平衡约束的数学规划。这些研究将引出我们下一个研究课题——鲁棒优化。