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.

凸优化是一个长期研究的基础研究领域。这也是一个非常热门的研究领域，最近已经开发了新的有效解决方案方法（例如内部点算法），在其中发现了各种实际重要性的应用。这项研究的目的是基于稳定的理论基础开发实际有效的方法，以解决凸优化问题和相关问题，这些问题涉及特定受限的凸面编程问题，积极的半定义编程问题以及单调互补性问题，从而有助于扩大工程应用程序的应用领域。该项目中开发的方法列出如下：（1）针对约束凸编程问题的顺序二次约束二次编程方法；（2）牛顿的正则化方法，用于最大程度地减少可能具有奇异黑森的凸功能；（3）单调互补性问题的近端方法中的主动集识别技术；（4）具有平衡约束的数学程序的迭代方法；（5）平滑牛顿方法的二阶互补问题；（6）用于二阶锥体互补性问题的矩阵分裂方法。此外，我们在非合作游戏中引入了一个新的平衡概念，并通过二阶锥体互补问题对其进行了研究。我们还研究了在不确定性下具有平衡限制的非线性半决赛编程问题和数学程序。这些研究将导致我们下一个强大优化的研究主题。