多目标(半)无限DC规划问题最优条件和对偶理论研究

(Semi-)Infinite programming is a very active research branch in applied mathemathics which has been widely applied in Engineering design, Optimal control, Information technology and Economic equilibrium. This project considers a special (semi-)infinite programming, that is, multiobjective (semi-)infinite DC programming in which every objective function and every constraint function can be decomposed into the difference of two convex functions. In this project, we will focus on the optimality conditions and duality theory for the multiobjective (semi-)infinite DC programming. Specifically, our contributions are threefolders: (1) Firstly, the Farkas lemma will be extended to the inequality systems including infinite DC functions. Then the optimal conditions will be established by the extended Farkas lemma which include both the optimally sufficient condition and the optimally necessary condition. The multiobjective (semi-)infinite linear or convex approximate programming to the original problem will also be proposed and the relationship of the optimal solutions to the primal problem and the approximate problem will also be analyzed. (2) Secondly, the dual problems will be given and the duality theories including the weak duality theory, strong duality theory and converse duality theorem will be presented. (3) Finally, as the applications of the above method, the optimality condtions and duality theories of the multiobjective (semi-)infinite fractional DC programming will be studied where every numerator and denominator in the fractions of the objective functions and the constraint functions are all DC functions, that is they can be decomposed into the difference of two convex functions.

(半)无限规划是应用数学中非常活跃的一个研究分支，在工程设计、最优控制、信息技术以及经济均衡等方面具有广泛的应用。本项目拟针对一类特殊的(半)无限规划问题- - 多目标(半)无限DC规划问题展开研究，试图对该类优化问题的最优条件和对偶问题进行深入研究，具体内容包括：（1）将Farkas引理推广至由无限多DC函数构成的不等式系统上，在此基础上讨论多目标(半)无限DC规划问题的最优必要和充分条件，建立原问题的多目标(半)无限线性规划或凸规划近似问题，并分析两者最优解(弱有效解或有效解)之间的关系；（2）研究原问题的对偶问题，讨论其对偶理论，重点构建弱对偶定理、强对偶定理和逆对偶定理；（3）在上述工作基础上研究多目标(半)无限分式DC规划问题(带有无限多DC函数约束，目标函数中每个分式的分子和分母均为DC函数的多目标规划问题)的最优条件和对偶理论。

1、针对一类特殊的多准则决策问题—半无限多目标DC规划问题，讨论了其最优性条件和对偶理论，并且将其和半无限线性规划问题进行比较，最为应用讨论了半无限多目标分式DC规划问题的最优性条件和对偶理论，这一成果发表在国际知名期刊《Journal of Global Optimization》上； 2. 针对一类特殊的多目标决策问题—带有线性约束的C-向量优化问题展开研究，讨论了这类问题的理论性质，求解算法以及其在供应链风险管理中的应用，这一成果发表在国际知名期刊《European Journal of Operational Research》 上；3. 多准则博弈理论是博弈论中的一个重要分支，多准则博弈能够更好的描述现实生活中的问题。当多准则博弈中的决策者需要在多个可能的权重中寻找最坏情况下的决策时，我们提出了一类新的鲁棒权重方法来处理这类问题。据此给出了鲁棒权重帕累托均衡解的概念，在一般条件下证明了此均衡解的存在性，这一成果发表在国际期刊《PLoS One》上；4. 当目标函数或者约束函数为非光滑时，多准则决策模型就为非光滑的，此时将广泛应用于求解单目标优化问题的拟牛顿算法推广至非光滑多目标优化问题上来，讨论了新方法的全局收敛性和局部超线性收敛性，并且给出了数值实验验证了新方法的有效性，这一成果发表在国际知名期刊《European Journal of Operational Research》上；5. 针对一类从多准则决策科学问题中抽象出的多目标DC规划问题，提出了一类新的近似点算法，讨论了新算法的收敛性质，最后将新方法用于讨论投资组合管理问题，这一成果发表在国际知名期刊《Journal of Optimization Theory and Applications》上.