一类非光滑优化问题的邻近交替方向法

Proximal alternating direction algorithm for a class of nonsmooth optimization

非光滑优化问题在现实生活中有着广泛应用.针对一类带有结构特征为两个连续凸函数与具有Lipschitz梯度的二次可微函数的和的无约束非光滑非凸优化问题,给出了一种邻近交替方向法,称之为二次上界逼近算法.该算法结合交替方向法与邻近点算法的思想,将上述优化问题转化为平行的子问题.在求解子问题的过程中,对目标函数中的光滑部分线性化,此时子问题被转化为凸优化问题.然后分别对两个凸优化子问题交替利用邻近点算法求解.基于以上思想,首先我们给出算法的伪代码,然后建立了算法收敛性的充分条件,最后证明在该条件下,算法产生迭代序列的每个极限点是原问题的临界点.

Nonsmooth optimization problem has a wide application in real life. For a class of unconstrained nonsmooth and nonconvex problem with structural function,which is combined two proper continuous convex functions and a second differentiable function with Lipschitz gradient,a kind of proximal alternating direction method is proposed,namely quadratic upper-bound approximation algorithm. Combined with alternating direction method and the idea of the proximal point algorithm,this algorithm is used to transform the above optimization problem into parallel sub-problems. In the process of solving the sub-problem,the smooth partial of the objective function should be linear,thus,the sub-problem is converted to a convex optimization problem. Then the two convex optimization sub-problems are solved by using proximal point algorithm respectively. Based on the above thoughts,we first give the pseudo code of the algorithm,then establish the sufficient condition of the convergence of the algorithm. Finally,it is proved that every limit point of the iterative sequence generated by the algorithm is the critical point of the original problem.