两类非凸约束优化问题的最优性条件及稳定性分析

By using theories of Banach space, convex analysis, nonliear functional analysis, nonsmooth analysis and variational analysis, we consider the optimality conditions and stability for DC (difference of two convex functions) infinite optimization problems and DC fractional programming. The objectives include the following: （1）the local and global ptimality conditions;(2)the closedness, lower semicontinuity, upper semicontinuity, Lipschitz continuity, Fréchet subdifferential, Mordukhovich subdifferential of the feasible set mapping, the solution set mapping and the optimal value function; (3)the computation of coderivative for the feasible set mapping; (4)the algorithms for solving DC optimization problems and for DC fractional programming; (5) applications to the best approximation and optimization problems with infinitely subsmooth functions, optimal control, semidefine programming, image restoration. This project can be recast into the fields of convex analysis, nonsmooth analysis, functional analysis, nonsmooth analysis, variational analysis and theory of functional approxiamtion. Hence, in view of application and theoretical development, our project is very meaningful and valuable.

本项目充分利用Banach空间几何理论、凸分析、非线性泛函分析、非光滑分析以及变分分析的有关理论，研究Banach空间中DC约束优化问题及DC分数规划的最优性条件和稳定性理论。主要研究内容包括：（1）DC约束优化问题和DC分数规划的局部和全局最优性条件；（2）DC约束优化问题和DC分数规划的可行集算子、最优解算子及最优值算子的闭性、下（上）半连续性、Lipschitz连续性及算子的 Fréchet次微分、Mordukhovich次微分；（3）DC约束优化问题和DC分数规划的可行集算子的协导数；（4）设计求解DC约束优化问题和DC分数规划的算法；（5）应用这些研究成果来研究带次光滑约束的逼近和优化问题、最优控制问题、半正定规划、图像恢复问题等。本项目是属于凸分析、泛函分析、非光滑分析、变分分析及函数逼近论等分支的交叉学科，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。

本项目充分利用Banach空间几何理论、凸分析、非线性泛函分析、非光滑分析以及变分分析的有关理论，研究Banach空间中DC规划、DC分数规划和DC复合优化问题的对偶理论、最优性条件和稳定性理论。主要研究内容及结果包括：（1）DC规划、DC分数规划、DC复合优化问题的弱对偶、强对偶、全对偶及零对偶等；（2）凸复合优化和DC复合优化问题问题的最优性条件；（3）DC约束优化问题的可行集算子、最优解算子及最优值算子的闭性、下（上）半连续性、Lipschitz连续性及算子的 Fréchet次微分、Mordukhovich次微分；（4）DC约束优化问题的可行集算子的协导数计算；（5）一类特殊优化问题--凸可行问题的求解算法及收敛性分析；（6）应用这些研究成果来研究带次光滑约束的逼近和优化问题、半正定规划等。本项目是属于凸分析、泛函分析、非光滑分析、变分分析及函数逼近论等分支的交叉学科，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。