求解消失约束数学规划问题的算法研究

Mathematical programs with vanishing constraints (MPVC) is a class of very important but numerically difficult optimization problems, which has many applications such as in the optimal topology design problems of mechanical structures and robot motion planning. In this project, we will mainly study the efficient algorithms for MPVC. The detailed research contents are listed as follows: (1) Based on the smoothing regularization about the partly vanishing constraints, some new smoothing regularization methods for solving the MPVC will be presented ,and the convergent properties of approximate KKT sequences generated by these methods will be discussed. (2) Based on the new nonsmooth reformulation of partly vanishing constraints which is proposed in (1), the lower order penalty function method and the augumented Lagrangian method to solve the MPVC will proposed, and their convergent properties and the feasibility about the accumulation point of the solution sequences will be discussed. (3) Based on the relaxation about the partly vanishing constraints, some new relaxation methods for solving the MPVC will be designed, and the convergent properties of corresponding approximate KKT sequences will be investigated. (4) Based on the new relaxation problem which is proposed in (3), the sequential quadratic programming method (SQP) to solve the MPVC will be constructed, and their global convergence properties will be established. For the above methods, we will report some numerical results to show their efficiency.

消失约束数学规划(MPVC)是一类非常重要但数值求解比较困难的优化问题，它在机械结构最优拓扑设计和机器人运动规划等领域具有广泛的应用。本项目旨在设计求解MPVC问题的有效算法，具体研究内容如下： (1) 提出基于部分消失约束光滑正则化的新光滑化正则化方法，并讨论新的光滑化正则化问题近似KKT点列的收敛性； (2) 基于研究内容(1)中给出的部分消失约束新的非光滑等价形式，提出求解MPVC问题的低阶罚函数法和增广拉格朗日方法，并讨论相应算法的收敛性质以及罚问题解序列的可行性； (3)设计基于部分消失约束松弛的新松弛方法，并讨论新的松弛问题近似KKT点列的收敛性； (4)基于研究内容(3)中给出的新松弛问题，构造求解MPVC问题的序列二次规划算法，并给出算法的全局收敛性分析；对以上算法，给出相应的数值结果，验证算法的有效性。

消失约束数学规划(MPVC)是一类非常重要但数值求解比较困难的优化问题，它在机械结构最优拓扑设计和机器人运动规划等领域具有广泛的应用。本项目主要研究MPVC 问题的若干理论和有效算法，其主要研究内容如下：.（1）给出了 Kanzow et al. (2013)提出的求解消失约束数学规划问题一个光滑化正则化方法的改进收敛性结果，主要证明了在VC-MFCQ和渐进非退化条件下，该光滑子问题解序列的强收敛性；另外，基于X.J. Chen(2012)的光滑技术, 给出了求解消失约束数学规划问题的一类光滑正则化方法.该类方法包括了Kanzow et al(2013)光滑化方法. 在比Kanzown et al方法条件弱的情况下，建立了该类方法的强收敛性。.（2）通过引入 MPVC 问题的 F-J 条件，给出了消失约束数学规划问题的几个新的约束规格，在此新的约束规格下讨论了该问题 MPVC型罚函数和l_1罚函数的精确性；.（3）基于消失约束的结构特征，给出了求解消失约束数学规划问题的部分增广拉格朗日方法和内点罚方法，并讨论了上述两种方法的收敛性；.（4）给出了消失约束数学规划问题的Wolfe对偶和Mond-Weir对偶的改进模型，该模型不涉及未知指标集的计算，克服了Mishra et al(2016)对偶模型的缺陷，并用实例解释了改进对偶模型的有效性；.（5）基于Logarithm-Exponential函数、Q. Li, D.-H. Li(Adv.Model. Optim. 13 (2011) 141–152)所给光滑函数等，给出了互补约束数学规划问题的三个光滑化方法，并在较弱条件下讨论了三个方法的收敛性。

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