复合函数型DC规划的最优性条件和鲁棒性分析

Many practical problems in the real world can be converted to a class of convex optimization problems or composite optimization problems or DC programming problems. In order to provide a unified new idea and new method for the study of these constrained optimization problems, we propose a new class of problems, namely, DC programming with composite functions. For the newly proposed problems, we intend to investigate the optimality conditions, stabiliity and robustness analysis by using theories of Banach space, convex analysis, nonsmooth analysis and variational analysis. The objectives include the following: (1) introduce some new weakly constraint qualifications and study these relationships; (2) duality and optimality conditions of DC programming with composite functions; (3) the closedness, lower semicontinuity, upper semicontinuity, Lipschitz continuity, generalized subdifferential of the feasible set mapping, the solution set mapping and the optimal value function; (4) Robust duality and robust optimality conditions for DC programming with composite functions with uncertainty data; (5) applications to semidefine programming, fractional programming, composite optimization problem, optimal control. This project can be recast into the fields of convex analysis, functional analysis, nonsmooth analysis and variational analysis. Hence, in view of application and theoretical development, our project is very meaningful and valuable.

鉴于现实世界中许多的实际问题都可以转化为一类凸优化问题或复合优化问题或DC规划问题，为给这些约束优化问题的研究提供一种统一的新思想和新方法，本项目提炼出一类新的复合函数型DC规划问题作为研究对象，充分利用Banach空间几何理论、凸分析、非光滑分析、变分分析的有关理论，研究复合函数型DC规划的最优性条件、稳定性及鲁棒性分析。主要研究内容包括：(1)引进新的弱性的约束规范条件并建立其内在联系；(2)对偶理论和最优解的特征刻画；(3)可行集算子、最优解算子及最优值算子的闭性、下（上）半连续性、Lipschitz连续性及广义次微分的计算；(4)数据不确定情形下的复合函数型DC规划的鲁棒对偶理论与鲁棒最优性条件刻画；(5)上述研究成果在半正定规划、分数规划、复合优化、最优控制等领域的应用。本项目是属于数学规划领域多分支的交叉，无论在理论上还是在应用前景上都有重要的研究价值和学术意义。

本项目以复合函数型DC规划问题为研究对象，充分利用Banach空间几何理论、凸分析、非光滑分析、变分分析的有关理论，研究了复合函数型DC规划的对偶理论、稳定性分析以及最优解和近似最优解的KKT类最优性条件。主要研究结果包括:(1) 凸复合优化问题、DC规划、DC复合优化问题、分式规划与其对偶问题之间的弱对偶、零对偶、强对偶、全对偶、Farkas引理以及近似弱对偶、近似零对偶、近似强对偶、近似全对偶、近似Farkas引理等; (2) 带复合函数的凸约束优化问题和DC约束优化问题的最优解和近似解的KKT类最优性条件和鞍点定理等; (3) DC锥约束优化问题的最优解集的特征刻画; (4) 带复合函数的DC无约束优化问题与其Fenchel对偶问题之间的弱对偶、零对偶、强对偶以及全对偶等; (5) 数据不确定情形下的复合函数型DC规划的鲁棒对偶、鲁棒可行解和鲁棒最优解的特征刻画以及局部和全局最优性条件成立的充分和必要条件(6) 两类广义凸约束优化问题的对偶理论与最优性条件; (7) 上述研究结论在凸规划、锥规划、复合优化、半无限规划的应用。本项目是属于数学规划领域多分支的交叉，无论在理论上还是在应用前景上都有重要的研究价值和学术意义。