求解不可分的非凸优化问题的广义惯性交替结构化邻近梯度下降算法

GENERAL INERTIAL ALTERNATING STRUCTURE-ADAPTED PROXIMAL GRADIENT DESCENT METHOD FOR SOLVING NONSEPARABLE NONCONVEX OPTIMIZATION PROBLEMS

本文考虑求解一类不可分的非凸非光滑优化问题,该问题的目标函数由如下两部分组成:关于全局变量不可分的正常下半连续双凸函数,与两个关于独立变量的无利普希茨连续梯度的非凸函数.本文提出广义的惯性交替结构化邻近梯度下降算法(general inertial alternating structure-adapted proximal gradient descent algorithm,简记为GIASAP算法),该算法框架不仅引入非线性邻近正则项与惯性加速技巧,同时采用常数步长与动态步长两种策略.本文证明了GIASAP算法O(1/k)的非渐近收敛率,以及当目标函数具有Kurdyka-Łojasiewicz性质时,由GIASAP算法生成的有界序列全局收敛到问题的驻点.最后,本文通过数值实验验证了算法的可行性与有效性.

This paper considers the nonseparable nonconvex nonsmooth minimization problem,whose objective function is the sum of a proper lower semicontinuous biconvex function of the entire variables,and two nonconvex functions of their private variables without the global Lipschitz gradient continuity.This paper develops a general inertial alternating structureadapted proximal gradient descent algorithm(GIASAP for short),which not only adopts nonlinear proximal regularization and inertial strategies,but also utilizes constant and dynamical step sizes.The worst case O(1/k)nonasymptotic convergence rate of GIASAP algorithm is established.Furthermore,the bounded sequence generated by GIASAP globally converges to a critical point under the condition that the objective function possesses the Kurdyka-Łojasiewicz property.In addition,numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.