基于三阶泰勒展开逼近目标函数的无约束最优化算法  

An Unconstrained Optimization Algorithm Based on the Three Orders Taylor Expansion Approaching Object Function

作　　者：侯小秋 HOU Xiaoqiu(School of Electronics and Controlling Engineering, Heilongjiang University of Science and Technology, Ha’erbin 150022, China)

机构地区：[1]黑龙江科技大学电气与控制工程学院,黑龙江哈尔滨150022

出　　处：《东莞理工学院学报》2021年第3期22-26,共5页Journal of Dongguan University of Technology

摘　　要：针对基于二阶泰勒展开逼近目标函数精度低的牛顿法优化问题,研究基于三阶泰勒展开逼近目标函数的最优化算法意义明确,算法归结为多元二次方程组的求解,应用非线性方程组的牛顿法求解,在目标函数中加入二次函数辅助项,提出两个改进的最优化算法,改进的算法1可保证牛顿法的雅可比矩阵非奇异,改进的算法2可保证牛顿法的雅可比矩阵正定,所提出的无约束最优化算法可推广到高阶泰勒展开情形,数值分析例验证了所提出的最优化算法的有效性。In view of the Newton method optimization problem based on the second order Taylor expansion approximation objective function with low precision,the significance of the optimization algorithm based on the third order Taylor expansion approximation objective function is clear.Two improved optimization algorithms are proposed by using Newton method of nonlinear equations and adding quadratic function auxiliary term to the objective function,the modified algorithm 1 guaranteeing the non-singular of Newton method’s Jacobi(an)matrix,and the modified algorithm 2 guaranteeing positive definite of Newton method’s Jacobi(an)matrix.the proposed unconstrained optimization algorithms can be extended to higher order Taylor expansion,and the results of numerical analysis verify its validity.

关 键 词：无约束最优化 泰勒展开 非线性方程组 非奇异矩阵 正定矩阵 

分 类 号：O242.23[理学—计算数学]