和目标函数具有相同局部极小点的打洞函数  

A Tunneling Function which Has the Same Local Minimizer of the Objective Function

作　　者：屈德强 李军祥[1] 尚有林[2] 潘龙博 QU Deqiang;LI Junxiang;SHANG Youlin;PAN Longbo(Business School,University of Shanghai for Science&Technology,Shanghai 200093;School of Mathematics and Statistics,Henan University of Science and Technology,Luoyang 471023)

机构地区：[1]上海理工大学管理学院,上海200093 [2]河南科技大学数学与统计学院,洛阳471023

出　　处：《工程数学学报》2024年第5期825-837,共13页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(72701130,71871144,12071112,11701150,11471102);河南省高等学校重点科研项目计划基础研究专项(20ZX001);上海理工大学大学生创新训练计划项目(XJ2023156).

摘　　要：打洞函数方法作为求解全局优化问题的一种有效方法,其跳出局部极值的能力深受打洞函数性质的影响。随着实际优化问题的复杂化,其对应的打洞函数形式更加复杂。因此,构造形式简单且性质良好的打洞函数是打洞函数方法的主要研究目标之一。为了提高打洞函数方法求解多峰函数的效率,提出了一个新型的打洞函数,其局部极小点不仅是比目标函数当前局部极小点更优的可行点,同时也是更优的局部极小点,即打洞函数和目标函数具有相同的局部极小点。于是,只需极小化打洞函数即可直接求得比目标函数更优的局部极小点。基于此特点,设计了一个新的打洞函数算法,该算法改进了传统打洞函数法的算法框架,克服了交替极小化目标函数和打洞函数的局面,有效地减少了局部寻优的次数,加快了全局寻优的速度。理论分析和数值实验验证了算法的可行性和有效性。Tunneling function method is an effective method for global optimization problems.Its ability to jump out of the local minimizer is deeply affected by the properties of the tunneling function.With the complexity of practical optimization problems,the corresponding forms of tunneling functions become more complex.Therefore,constructing tunneling functions with simple form and good properties is one of the main research objectives of the tunneling function method.In order to improve the efficiency of the tunneling function method for solving multi-modal functions,a new tunneling function is constructed.Its minimizers are not only the better feasible points of the objective function than the current local minimizer,but also the better local minimizers,i.e.,the tunneling function and the objective function have the same local minimizers.Thus,the better local minimizer of the objective function can be obtained directly by minimizing the tunneling function.Based on this feature,a new tunneling function algorithm is designed.It changes the frame of conventional tunneling function methods that objective function and tunneling function are minimized alternately,and can effectively reduce the iterations of local optimization and accelerate the speed of global optimization.Theoretical analysis and numerical experiments exhibit the feasibility and effectiveness of the algorithm.

关 键 词：全局最优化 打洞函数 局部寻优 局部极小点 

分 类 号：O221.2[理学—运筹学与控制论]