带固定步长的非单调信赖域三点步长梯度算法  

A NON-MONOTONE TRUST RIGION THREE-POINT STEP SIZE GRDIENT METHOD WITH FIXED STEP SIZE

作　　者：孙清滢 韩少华 桑兆阳 Sun Qingying;Han Shaohua;Sang Zhaoyang(College of Science,China University of Petroleum(East China),Qingdao 266580)

机构地区：[1]中国石油大学(华东)理学院,青岛266580

出　　处：《高等学校计算数学学报》2024年第3期193-208,共16页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(51974343)资助项目

摘　　要：1引言考虑无约束最优化问题:■f(x),(1.1)其中f(x):R^(n)→R^(1)是连续可微函数.众所周知,信赖域算法[1-6]是求解问题(1.1)的重要算法,具有较强的全局收敛性质和较快的局部收敛速度.信赖域子问题的求解是信赖域算法的关键问题,算法的工作量主要是子问题的求解.Based on point difference vectors and gradient difference vectors over the most recent three iterations,together with the Taylor’s theorem,two forms of the quasi-Newton equation at the recent iteration are given.Using the method of least squares,we derive the three-point step size gradient method by the two forms of the quasi-Newton equation.By using the step of the three-point step size gradient method,we derive a simple quadratic model.Based on the simple quadratic model,and combining Gu and Mo’s non-monotone strategy with a fixed step size,we propose a non-monotone trust region three-point step size gradient method for unconstrained optimization problems.When a trial step is not accepted,the method does not need to resolve the sub-problem but generates an iterative point whose step length is defined by a formula.Under certain conditions,the global convergence properties of our new method are proved.Numerical results show that the new algorithm is efficient.Because of its simplicity,multi-point information,robustness,low memory requirement and only first order information being used,the new method is very suitable for solving large-scale optimization problems.

关 键 词：信赖域算法 连续可微函数 信赖域子问题 全局收敛 固定步长 局部收敛 非单调 梯度算法 

分 类 号：O221.2[理学—运筹学与控制论]