一类非线性三层规划问题的罚函数方法

A Penalty Function Method for Solving a Class of Nonlinear Trilevel Programming Problem

文章研究了一类结构为非线性-线性-线性三:层规划问题的求解方法.首先,基于下层问题的Karush-Kuhn-Tucker (K-K-T)最优性条件,将该类非线性三层规划问题转化为具有互补约束的非线性二层规划,同时将下层问题的互补约束作为罚项添加到上层目标;然后,再次利用下层问题的K-K-T最优性条件将非线性二层规划转化为非线性单层规划,并再次将得到的互补约束作为上层目标的罚项,构造了该类非线性三层规划问题的罚问题.通过对罚问题性质的分析,得到了该类非线性三层规划问题最优解的必要条件,并设计了罚函数算法.数值结果表明所设计的罚函数算法是可行、有效的.

In this paper,we mainly focus on the method for solving a class of trilevel programming problem,where the objective functions of the upper,middle,and lower levels are nonlinear,linear,and linear,respectively.Firstly,based on the KarushKuhn-Tucker(K-K-T) optimality condition of the lower level problem,the original problem is transformed into a nonlinear bilevel programming problem with complementary constraints.Subsequently,the complementary constraints of the lower level problem are added to the upper level objective as penalties.Then,we use the KK-T optimality condition of the inside problem to transform the nonlinear bilevel programming problem into a nonlinear single-level programming problem,and the obtained complementary constraints are again used as the penalty term for the upper level objective.Therefore,a penalized problem for the nonlinear trilevel programming problem is constructed.Through the analysis of the characteristics of the penalized problem,the necessary conditions for the optimal solution of the nonlinear trilevel programming problem are obtained,and the penalty function algorithm is designed.The numerical results show that the proposed penalty function algorithm is feasible and effective.