Theory and computational methods of robust optimization

本课题在对一类鲁棒优化问题进行理论分析的基础上，研究了一类鲁棒优化问题的高效算法。这类问题一般表示为以不确定性为参数的最小-最大问题。本文首先研究了(1)基于切割平面策略的一些数值算法的不精确实现，最后提出了(2)一个双边切割平面框架作为最终解的形式，对于一类在测度空间中表述的优化问题，该框架具有全局收敛性。此外，我们还尝试了一些数值框架的改进，使其具有更好的局部收敛性，从而更有效地实现。本研究项目的实际方面包括(3)在半无限规划特别是数字滤波器设计方面的一些应用，(4)理论分析了所开发的框架在决策变量限制为一类绝对连续测度的特殊情况下的渐近行为，以及(5)在具有状态变量不等式约束的最优控制问题上的一些应用。作为相关课题，我们还研究了(6)一类基于连续下降方法的路径跟踪算法和(7)常微分方程刚性系统积分的数值分析。

This research project aims at developing efficient algorithms for a class of robust optimization problem based on its theoretical analysis. Such a class of problem is represented in general as a min-max problem involving uncertainty as parameters. We first studied (1) inexact implementations of some numerical algorithms based on the cutting plane strategy, and finally proposed (2) a bilateral cutting plane framework as an ultimate solution form, whose global convergence is shown for a class of optimization problem formulated in a measure space. Furthermore we tried some refinements of the numerical framework towards its more efficient implementation with better local convergence properties. The practical side of this research project includes (3) some applications to semi-infinite programming especially for digital filter design, (4) a theoretical analysis of the asymptotic behavior of the developed framework when applied to a special case whose decision variable is limited to a class of absolutely continuous measure, and (5) some applications to optimal control problems with state variable inequality constraints. We also investigated, as related topics, (6) a type of path-following algorithm based on continuous descent methods and (7) a numerical analysis of the integration of stiff systems of ordinary differential equations.