参数规划的最优值函数优化问题的理论研究与算法实现

The theory of mathematical program with equilibrium constraint has important value in solving computational mechanics with developping internal variable. The functions in mathematical program with equilibrium constraint models are all the optimal values or optimal solutions in parametric optimization or variational inequalities. Therefore, the investigation of theory problems and efficient algorithms for the optimization problems of optimal value functions in parametric programming is very important. The project focuses on the differential theory and efficient algorithms of the optimization problems of optimal value functions in parametric programming which have smooth structures. The research content includes the differential decomposition theory, the second order expansion theory and the U-Lagrangian proximal point theory of the optimization problems of optimal value functions in parametric programming. We construct numerical algorithms with super-linear convergence rate of the optimization problems of optimal value functions in parametric programming by using the tools of variational analysis and combining the theory of bundle method, semi-smooth Newton method and the differential decomposition theory.The expected theoretical results will play an active role in promoting the development of theoretical and application research in nonsmooth optimization. The application field of nonsmooth optimization will be broadened by the achievement in the research on algorithms.

均衡约束规划的理论对于求解具有发展型内变量的计算力学问题具有重要价值，均衡约束规划模型中的函数均是参数优化或变分不等式的最优值或最优解。因此，参数规划的最优值函数优化问题的理论以及算法的研究对于计算力学问题的解决至关重要。本项目重点研究具有光滑结构的参数规划的最优值函数优化问题的微分理论与算法。研究内容包括参数规划最优值函数优化问题中函数的微分分解理论，二阶展开理论与U-Lagrange的邻近点理论。以变分分析为工具，结合束方法和非光滑Newton方法的理论与所获得的微分分解理论，构造参数规划最优值函数优化问题的具有超线性收敛速度的数值算法。预期获得的理论成果对非光滑优化的理论与应用研究起推动作用，取得的算法成果将拓宽非光滑优化的应用领域。

参数规划的最优值函数优化问题的理论与算法的研究对于计算力学问题至关重要。本项目研究了具有光滑结构的参数规划的最优值函数优化问题的VU－分解理论与算法。研究内容包括参数规划的最优值函数优化问题的微分性质，利用微分结构得到了两个直交的子空间V和U，讨论了快速轨道的具体表达式，结合半光滑Newton方法及束方法得到了具有超线性收敛速度的算法。最后，利用Matlab软件验证了算法的有效性。