光滑神经网络解决非李普西茨优化问题的研究

为寻求满足约束条件的优化问题的最优解,针对目标函数是非李普西茨函数,可行域由线性不等式或非线性不等式约束函数组成的区域的优化问题,构造了一种光滑神经网络模型。此模型通过引进光滑逼近技术将目标函数由非光滑函数转换成相应的光滑函数以及结合惩罚函数方法所构造而成。通过详细的理论分析证明了不论初始点在可行域内还是在可行域外,光滑神经网络的解都具有一致有界性和全局性,以及光滑神经网络的任意聚点都是原始优化问题的稳定点等结论。最后通过几个简单的仿真实验证明了理论的正确性。

Solving non-Lipschitz optimization problems by smoothing neural networks

In order to seek to optimal solution satisfying the necessary conditions of optimality, aiming at the optimization problems that objective functions are non Lipschitz and the feasible region consists of linear inequality or nonlinear inequality, we design a new smooth neural network by the penalty function method and the smoothing approximate techniques which convert non smoothing objective functions into smoothing functions. Detailed theoretical analysis proves the uniform boundedness and globality of the solutions to smooth neural networks, regardless of the initial points inside or outside of the feasible domain. Moreover, any accumulation point of the solutions of to smooth neural networks is a stationary point of the optimization promble. Numerical examples also demonstrate the effectiveness of the method.