一类非线性三层规划问题的求解算法  

An algorithm for solving a class of nonlinear trilevel programming problem

作　　者：吕一兵[1] 吴伟 LYU Yibing;WU Wei(School of Information and Mathematics,Yangtze University,Jingzhou 434023,Hubei)

机构地区：[1]长江大学信息与数学学院,湖北荆州434023

出　　处：《长江大学学报(自然科学版)》2025年第1期103-110,共8页Journal of Yangtze University(Natural Science Edition)

基　　金：国家自然科学基金项目“三层规划问题的算法设计与应用研究”(12271061)。

摘　　要：针对一类结构为非线性-线性-线性的三层规划问题,提出了一种求解方法。首先,将下层问题的对偶间隙作为上层目标的罚项,将该类非线性三层规划问题转化为一类二层规划问题;然后,再次取所得到的二层规划问题的下层问题的对偶间隙为上层目标的罚项,构造了该类非线性三层规划问题的罚问题。通过引入两次对偶变换和罚函数方法,将原问题转化为可求解的单层非线性规划问题。证明了罚问题最优解与原问题最优解的等价性,并给出了最优解的必要条件。基于此设计了一种算法,通过数值实验验证了该算法的可行性。In this paper,a method is proposed for a class of nonlinear-linear-linear trilevel programming problems.Firstly,the duality gap of the lower level problem is employed as the penalty term of the upper level objective,converting the nonlinear trilevel programming problem into a type of bilevel programming problem.Then,the duality gap of the lower level problem of the obtained bilevel programming problem is taken as the penalty term of the upper objective function,and the penalty problem of this kind of nonlinear trilevel programming problem is constructed.By incorporating two dual transformations and penalty function methodologies,the original problem is reformulated into a tractable single-layer nonlinear programming problem.The equivalence between the optimal solutions of the penalty problem and the original problem is rigorously established,and the necessary conditions for optimality are delineated.Based on this,an algorithm is designed and its effectiveness is verified by numerical experiments.

关 键 词：非线性三层规划 罚函数 对偶间隙 最优解 

分 类 号：O221[理学—运筹学与控制论]