多目标鲁棒凸规划问题：理论、算法和应用研究

Robust optimization approach has been extensively studied in the scalar optimization problems with uncertain parameters. But there are few researches about robust optimization problems involved with multiple uncertain objectives which makes it more difficult. In this project, we focus on the robust multiobjective convex programming and study its optimality conditions, algorithms and applications. We think that our primary contributions of this project are as follows: (1) Firstly, we study the optimality conditions of the robust Pareto optimum definitions for the robust multiobjective programming which include the optimality conditions for the linear robust multiobjective programming and the general robust multiobjective convex programming; (2) Secondly, we present a nonscalarization method to transform the robust multiobjective programming into robust scalar programming which makes it possible to bridge the gap between the robust scalar optimization and this setting. we formulate the equivalent problems for the linear robust multiobjective programming and put forward the approximate problem for the general robust multiobjective convex programming, respectively. Design the algorithms for solving the linear robust multiobjective programming and the general convex robust multiobjective programming, respectively; (3) Finally, the new method will be applied to the portfolio management problem and the supply chain management problem, respectively. The effectively numerical software will be presented. The comparisons to the methods based on stochastic programming will also be conducted.

鲁棒优化方法常用来解决不确定单目标规划问题，而对其在不确定多目标规划问题方面的应用研究却比较少，这主要是因为同时有多个目标的不确定性使得鲁棒优化方法的应用变得更加复杂。本项目拟针对多目标鲁棒凸规划问题展开研究，试图对该类规划问题的最优条件、算法和应用进行深入研究，具体内容包括：（1）研究基于鲁棒帕累托(Pareto)最优定义的最优条件，包括研究多目标鲁棒线性规划问题和一般多目标鲁棒凸规划问题的最优条件；（2）研究基于非标量化方法的多目标鲁棒凸规划问题的求解算法，包括给出不同支撑集合下的多目标鲁棒线性规划问题的对应等价问题和一般多目标鲁棒凸规划问题的近似问题，设计求解这两类问题的计算方法；（3）对不确定多目标投资组合管理及不确定情况下的供应链管理问题，探讨其自身结构及模型特征，应用新方法求解并编制实用有效的计算机软件，并将新方法与基于随机规划的方法进行比较。

鲁棒优化方法常用来解决不确定单目标规划问题，而对其在不确定多目标规划问题方面的应用研究却比较少，这主要是因为同时有多个目标的不确定性使得鲁棒优化方法的应用变得更加复杂。本项目拟针对多目标鲁棒凸规划问题展开研究，试图对该类规划问题的最优条件、算法和应用进行深入研究，具体内容包括：（1）研究基于鲁棒帕累托(Pareto)最优定义的最优条件，包括研究多目标鲁棒线性规划问题和一般多目标鲁棒凸规划问题的最优条件；（2）研究基于非标量化方法的多目标鲁棒凸规划问题的求解算法，包括给出不同支撑集合下的多目标鲁棒线性规划问题的对应等价问题和一般多目标鲁棒凸规划问题的近似问题，设计求解这两类问题的计算方法；（3）对不确定多目标投资组合管理及不确定情况下的供应链管理问题，探讨其自身结构及模型特征，并将新方法与基于随机规划的方法进行比较。