一类分布鲁棒优化问题的线性化方法及其应用  

A Linearizing Method for Distributionally Robust Optimization Problem and Applications

作　　者：纪颖[1] 李一军[2] 芦鹏宇[2] 周勇[2] 

机构地区：[1]哈尔滨工业大学基础与交叉科研院,哈尔滨150080 [2]哈尔滨工业大学管理学院,哈尔滨150080

出　　处：《应用数学学报》2013年第2期376-384,共9页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金重点(No.71031003);国家自然科学基金(No.71201040;71071041;11201099);黑龙江省博士后启动基金(No.LBH-Q11118)资助项目

摘　　要：本文考虑一类特殊的极大极小化问题,即分布鲁棒优化问题.这类优化方法是不同于随机规划和鲁棒优化的一类方法,在这类问题中,不确定变量的概率分布往往是不能精确得知的,只知道概率分布所满足的一些条件,比如一次信息、二次信息以及支撑集合信息等.如此分布鲁棒优化问题便是寻求在所有满足条件的分布中找寻满足最坏可能分布的解.一般情况下,这类优化问题的求解都是NP难的.本文考虑一类简单的情形,即考虑不确定变量的概率分布只满足一次信息、支撑集合信息以及仿射一次信息,通过应用半无限规划问题的对偶性,本文指出这类分布鲁棒优化问题等价于线性规划问题,从而原分布鲁棒优化问题可以应用现成的求解线性规划的方法进行求解.为验证方法的有效性,本文将新方法应用于解决不确定条件下含有交易费用的利率管理问题.This paper considers a special Max-Min problem that is the distributionally robust optimization problem which is different from stochastic programming and robust op- timization. In this kind of problem, the uncertain variable＇s probability distribution often can＇t be accurately acquired but with some known information about its probability, such as the first-order, second-order and support set information. Then the distributionally robust optimization problem is to find the worst-case solution under all possible distributions. In general to find the solution for this problem is NP hard. In this paper, we suppose a special case where the decision-maker gets across some parts of information about the uncertain distributions, for example the first-order, support set and affine first-order information. By applying the duality of the semi-infinite programming, the distributionally robust optimiza-tion problem can be equivalently reformulated as a linear optimization problem, and then it can be solved by some well-established linear programming approach. To verify the effec-tiveness of the method, we discuss an applications to portfolio management problem with transaction costs.

关 键 词：极大极小化问题 分布鲁棒优化 半无限规划 不确定条件 利率管理 

分 类 号：TP309[自动化与计算机技术—计算机系统结构]