光滑神经网络解决一类线性约束的非Lipschitz优化问题

Smooth Neural Network for Solving a Class of Non-lipschitz Optimization Problems withLinear Constraints

带有线性不等式约束的非光滑非优化问题被广泛应用于稀疏优化,具有重要的研究价值.为了解决这类问题,提出了一种基于光滑化和微分包含理论的神经网络模型.通过理论分析,证明了所提神经网络的状态解全局存在,轨迹能够在有限时间进入可行域并永驻其中,且任何聚点都是目标优化问题的广义稳定点.最后给出数值实验和图像复原实验验证神经网络在理论和应用中的有效性.与现有神经网络相比,它具有以下优势:初始点可以任意选取;避免计算精确罚因子;无需求解复杂的投影算子.

The nonsmooth non-Lipschitz optimization problem with linear inequality constraints is widely used in sparse optimization and has important research value.In order to solve such problems,a neural network model based on smoothing and differential inclusion theory is proposed.Through theoretical analysis,we prove that the state solution of the proposed neural network exists globally,the trajectory can enter the feasible region in a finite time and stay in it forever,and any accumulation point is the generalized stationary point of the objective optimization problem.Finally,some numerical experiments and image restoration experiments are presented to verify the effectiveness of the neural network in theory and application.Compared with some existing neural networks,it has the following advantages:the initial point can be selected arbitrarily;avoiding the calculation of the exact penalty factor;it does not need to solve the complicated projection operator.