光滑化方法及其应用概述

An Overview of Smoothing Methods and Their Applications

互补问题、非线性约束优化问题的罚函数和极小极大函数等都是非常典型的非光滑问题(函数)。利用二次函数法、密度函数法及最大熵函数法等常用的光滑化方法,将非光滑问题(函数)转化为光滑化问题(函数),并给出其在智能电网实时电价定价中的应用。将智能电网实时定价的社会福利最大化模型求解转化为一个含有互补问题的方程组求解。根据互补理论和经济学含义,可以将互补问题转化为包含影子价格的非光滑方程组;利用光滑化方法,得到包含影子价格的光滑方程组;应用拟牛顿法、雅可比光滑化牛顿法等算法求解光滑方程组,得到最优用电量、最优生产电量和电力价格。

Complementary problems, penalty functions of nonlinear constrained optimization problems and minimax functions are all typical nonsmooth problems(functions). The nonsmooth problems(functions) are transformed into smoothing problems(functions) by using the usual smoothing methods, such as quadratic function method, density function method and maximum entropy function method. And the application of smoothing methods in real-time pricing for smart grid is given. The social welfare maximization model of real-time pricing for smart grid is transformed into a set of equations with a complementary problem. According to the complementary theory and economic meaning, the complementary problem can be converted into nonsmooth equations with the shadow price. Then the smooth equations with shadow price is obtained by using the smoothing methods. At the end, the optimal electric power consumption, the optimal electric power production and the electricity price can be easily given with quasi-newton method and Jacobi smoothing Newton method.