关于二阶锥互补约束数学规划问题的约束规范和算法研究

Second-order cone programming problem (SOCP) has lots of applications such as antenna array weight design, finite response impulse filter design, and portfolio optimization. However, little study has been done on the mathematical program with second-order cone complementarity constrains (MPSOCC). The developed mathematical program with equilibrium constraints can be regarded as a special case of the MPSOCC. In addition, if a bilevel programming problem contains a convex SOCP as lower level program, it can be formulated as an MPSOCC. This project is to study MPSOCC constraint qualification and algorithms. In particular, we will study Robinson constraint qualification and augmented Lagrangian method for MPSOCC. A local minimizer of MPSOCC must be M-stationary under MPSOCC-Robinson constraint qualification. We show that the limiting point of a sequence of optimal solution of penalty problems is stationarity of the original MPSOCC if the limiting point satisfy the suitable condition and design appropriate algorithm.

二阶锥规划问题在天线阵列权重的设计，有限响应脉冲(FIR) 滤波器的设计，组合优化和磁屏蔽设计优化有广泛的应用，但是很少有关于二阶锥互补约束数学规划问题（MPSOCC）的研究。MPSOCC可以看成是比均衡约束数学规划问题更一般的问题，双层规划的下层是凸的二阶锥规划时双层规划可以转化为MPSOCC。本项目研究MPSOCC的约束规范和算法，特别地，我们将研究MPSOCC的Robinson约束规范和求解MPSOCC的增广拉格朗日方法，证明MPSOCC的局部最优解在MPSOCC-Robinson约束规范条件下是M-稳定点的结论，建立罚问题的局部最优解的聚点在适当的约束规范条件下是MPSOCC的稳定点的结论，并设计相应算法。

二阶锥规划问题在天线阵列权重的设计，有限响应脉冲(FIR) 滤波器的设计，组合优化和磁屏蔽设计优化有广泛的应用，但是很少有关于二阶锥互补约束数学规划问题（MPSOCC）的研究。MPSOCC可以看成是比均衡约束数学规划问题更一般的问题，双层规划的下层是凸的二阶锥规划时双层规划可以转化为MPSOCC。本项目研究了求解MPSOCC的增广拉格朗日方法，证明了罚问题的局部最优解的聚点在MPSOCC-非退化条件条件下是原MPSOCC的C-稳定点的结论，罚问题的全局最优解的聚点在一定的条件下是原MPSOCC问题的全局最优解。