求解非凸优化问题的近似交替方向乘子法

Approximate ADMM for Non-Convex Optimization Problems

考虑有界约束上具有可分结构的非凸优化问题,提出了一种基于ADMM的新算法P-ADMM(即近似ADMM).在基于ADMM的框架下,P-ADMM在解决有界约束上的非凸子问题时,采用梯度投影,以此简化非凸子问题的求解,降低运算成本,并且通过引入一个“平滑的”(即指数加权)原始迭代序列,在每次迭代时,向增广拉格朗日函数中增加一个以平滑的原始迭代为中心的近似二次项,使所得到的近似增广拉格朗日函数在每次迭代时被不精确地最小化,在保证算法收敛性的同时也能够提升算法的收敛速度.数值实验表明,该算法可有效应用于求解一类非凸的船舶分布式能源管理问题.

In this paper,a new algorithm P-ADMM(approximate ADMM) based on ADMM is proposed for non-convex optimization problems with separable structures under bounded constraints.In the framework based on ADMM,P-ADMM adopts gradient projection to solve non-convex sub-problems with bounded constraints,so as to simplify the solving of non-convex sub-problems and reduce the cost of operation.Moreover,by introducing a “smooth”(exponential weighted) original iteration sequence,and at each iteration,by adding an approximate quadratic term centered on smooth original iteration to the augmented Lagrange function,the obtained approximate augmented Lagrange function is inaccurately minimized in each iteration,which can not only ensure the convergence of the algorithm but also improve the convergence speed of the algorithm.Numerical experiments show that this algorithm can be effectively applied to a class of non-convex ship distributed energy management problem.