开关约束数学规划(MPSC)是一类非常重要但数值求解比较困难的优化问题，它在开关约束最优控制、生产计划、投资组合优化、信号处理中的压缩感知和回归中的子集选择等问题中具有广泛的应用。本项目旨在研究MPSC问题的理论与算法，具体研究内容如下：.(1) 基于MPSC的加强Fritz John条件，提出几个新的MPSC型约束规格，并在此类约束规格下讨论MPSC型罚函数和MPSC问题的l_1罚函数的精确罚性质，设计求解MPSC问题的增广拉格朗日方法并在此类约束规格下分析其收敛性；.(2) 设计求解MPSC问题的新型松弛方法，并讨论该松弛方法在较弱约束规格下的收敛性；.(3) 构造求解MPSC问题的SQP方法，并给出该算法在Q_M稳定性意义下的全局收敛性质分析；.对以上算法，给出相应的数值结果，验证算法的有效性。

Mathematical programs with switching constraints (MPSC) are a class of very important but numerically difficult optimization problems, which have many applications such as in the switching constrained optimal control, production planning, portfolio optimization, compressed sensing in signal processing and subset selection in regression. In this project, we will mainly study some theories and algorithms for the MPSC. The detailed research contents are listed as follows: .(1) Based on the enhanced Fritz John conditions, we will introduce some MPSC type constraint qualifications, and discuss the exact penalty properties about the MPSC type penalty function and l_1 penalty function for the MPSC under such constraint qualifications. We also will propose the augumented Lagrangian method to solve the MPSC, and discuss its convergent properties under such constraint qualifications. .(2) We will design some new relaxation methods for solving the MPSC and investigate their convergent properties under the weak constraint qualification. .(3) We will construct the SQP method to solve the MPSC, and establish its global convergence property based on the Q_M stationarity. . For the above methods, we will report some numerical results to show their efficiency.