多面体约束非光滑复合函数的序列有效集方法

对带多面体约束的非光滑复合函数问题的求解进行了研究。针对非光滑复合函数问题，首先，构造光滑函数来逼近非光滑目标函数，通过求解光滑近似问题来达到求解原问题的目的。在此基础上，考虑多面体约束的特殊结构，运用序列二次规划算法的思想，利用有效集策略，通过逐次求解一系列仅含等式约束的二次规划问题来逼近搜索方向的最优解，再通过线搜索求得步长，进而得到下一步的迭代点。最后，从理论上证明了算法的全局收敛性，并进行了初步的数值实验。将该算法与光滑序列投影收缩算法作对比，结果表明，该算法在迭代次数和计算时间上都有一定的优势。

A sequential active set algorithm for nonsmooth composite functions with polyhedral constraints

The solution of nonsmooth composite function problem with polyhedral constraints was studied. For the problem of nonsmooth composite function, a smooth function was firstly constructed to approximate the nonsmooth objective function, and the purpose of solving the original problem was then achieved by handling the smooth approximation problem. On this basis, considering the special structure of polyhedral constraints, the idea of sequential quadratic programming algorithm was adopted. A quadratic programming subproblem was solved to obtain the search direction. The active set strategy was adopted by the solution of quadratic programming to approach the optimal solution of the search direction by solving a series of quadratic programming problems with equality constraints step by step. After that, the step size was obtained by line search. The next iteration point was then obtained. Finally, the global convergence of the algorithm was proved theoretically. A preliminary numerical experiment was carried out to compare the algorithm with the smoothed sequential projection shrinkage algorithm. The results show that the algorithm has certain advantages not only in iteration times but also in computing time.