The Institute of Statistical Mathematics

本研究项目的目的是开发一套适用于广义半无限规划(SIP)的高效数值方法。本文提出的数值方法包括：(1)基于对偶理论的求解凸SIP问题的算法；(2)基于序列二次规划框架下的局部约简的求解非线性SIP问题的算法；(3)基于不可微优化理论的求解非线性SIP问题的算法。实现了(4)基于割平面策略的近似算法和(5)基于割平面和局部约简的两阶段算法。作为进一步的研究方向，我们研究了(6)路径跟踪算法。基于连续下降法和相关的刚性常微分方程组，以及(7)基于半定规划松弛的数值方法。其中一些数值算法已经实现并应用于最优过滤器设计、全局优化、最小-最大规划、无限对策和最优控制问题，这些算法现在正在编译成一个用MatLab环境编写的软件包，很快就可以通过我们的网站获得。

The purpose of this research project is to develop efficient numerical methods, with a software package, for generalized semi-infinite programming (SIP). Numerical methods developed in this project include (1) an algorithm based on duality theory for solving convex SIP problems, (2) an algorithm based on local reduction within a framework of sequential quadratic programming for solving nonlinear SIP problems, (3) an algorithm based on nondifferentiable optimization theory for solving nonlinear SIP problems. We have also implemented (4) an approximation algorithm based on cutting plane strategy and (5) a two-phase algorithm based on cutting plane and local reduction. Also as further research directions, we have investigated (6) a path-following algorithm. based on continuous descent methods, together with related stiff systems of ordinary differential equations, and (7) numerical methods based on semi definite programming relaxations. Some of these numerical algorithms have been implemented and applied to problems of optimal filter design, global optimization, min-max programming, infinite games and optimal control, and these are now being compiled into a software package written in the Matlab environment, which will soon become available through our web site.