一类不可微二次规划逆问题

本文求解了一类二次规划的逆问题，具体为目标函数是矩阵谱范数与向量无穷范数之和的最小化问题.首先将该问题转化为目标函数可分离变量的凸优化问题，提出用G-ADMM法求解.并结合奇异值阈值算法，Moreau-Yosida正则化算法，matlab优化工具箱的quadprog函数来精确求解相应的子问题.而对于其中一个子问题的精确求解过程中发现其仍是目标函数可分离变量的凸优化问题，由于其变量都是矩阵，所以采用适合多个矩阵变量的交替方向法求解，通过引入新的变量，使其每个子问题的解都具有显示表达式.最后给出采用的G-ADMM法求解本文问题的数值实验.数据表明，本文所采用的方法能够高效快速地解决该二次规划逆问题.

A TYPE OF NON-DIFFERENTIABLE INVERSE QUADRATIC PROGRAMMING PROBLEMS

In this paper, a type of inverse quadratic programming problem is considered, which is a minimization problem of the sum of the matrix spectrum norm and the vector infinite norm. Firstly, the problem is transformed into a convex optimization problem with the objective function separable, and G-ADMM method is proposed to solve it. Then, we use the singular value threshold method, Moreau-Yosida regularization algorithm and the quadprog function in MATLAB optimization toolbox to solve the corresponding subproblem accurately. It is found that one subproblem is still a convex optimization problem with separable variables objective function. Because its variables are all matrices, so we adopt the alternative direction method suitable for multiple matrix variables to solve it. By introducing a new variable, we obtain that the solution of each subproblem has a display expression. Finally, the convergence analysis and numerical experiments of the G-ADMM method are given. The numerical experiments show that this method can solve the inverse quadratic programming problem efficiently and quickly.