APenalty Function Method for Solving a Class of Nonlinear Trilevel Programming Problem
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 Karush-Kuhn-Tucker(K-K-T)optimality condition of the lower level problem,the original problem is transformed into a nonlinear bilevel programming problem with comple-mentary constraints.Subsequently,the complementary constraints of the lower level problem are added to the upper level objective as penalties.Then,we use the K-K-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.
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