Research On Penalty Function Methods For Two Types Of Trilevel Programming Problems

Optimization theory is the theory and method of optimal design,optimal control and optimal management of the system.It is widely used in many fields such as national defense construction,engineering design,transportation,production management and so on.We call the optimization problem of large-scale hierarchical structures emerging in these numerous fields as multilevel programming.In recent years,many scholars have done a lot of research on multilevel programming and made significant achievements,but with the development of economy and society,the scale of the problem has become more and more complex and huge,no longer limited to simple bilevel programming,but involves three levels or even more than three levels,therefore,it is of great theoretical significance and application value to study the properties and solution methods of trilevel programming.This paper mainly studies the penalty function approach for two types of trilevel programming problems.The main work is as follows:(1)A penalty function method for solving the generalized linear trilevel programming problem is proposed.Firstly,based on the Kuhn-Tucker optimality condition of the lower level problem,the linear trilevel programming is transformed into a bilevel programming with complementary constraints.Then,the complementary constraints are attached to the upper level target as a penalty to obtain a bilevel programming problem with a penalty.Secondly,for the punitive bilevel programming problem,the Kuhn-Tucker optimality condition is used again,and the complementary constraints continue to be attached as punishment to the upper level target,forming a holistic penalty problem.Finally,by analyzing the optimal solution characteristics of the overall penalty problem,a penalty function algorithm is proposed,and the example results also show the feasibility and effectiveness of the algorithm.(2)A class of trilevel programming problems with a nonlinear-linear-linear structure is studied.Firstly,the Kuhn-Tucker optimality condition is used to transform the nonlinear trilevel programming into a nonlinear bilevel programming with complementary constraints,and the complementary constraints of the lower problems are attached to the upper level target as penalties.Then,with the help of the Kuhn-Tucker conditions again,it is transformed into a nonlinear single-level programming,and the new complementary constraints are used as a penalty term to construct a holistic penalty problem.Through the analysis of the nature of the penalty problem,the necessary conditions for the optimal solution are obtained,and the penalty function algorithm is designed on this basis.Numerical results show that the algorithm is feasible and effective.