A solving approach for the nonlinear trilevel programming problem with linear programming problem in the lower level

In this paper, we mainly focus on the solving approach for a class of nonlinear trilevel programming problem. Firstly, based on the Karush–Kuhn–Tucker (K–K–T) optimality conditions of the lower level problem, we transform the nonlinear trilevel programming problem into the nonlinear bilevel programming problem with complementary constraints. The complementary constraints of the lower level problem are added to the upper level objective as the penalties. Secondly, for the nonlinear bilevel programming problem, the lower bounding problem is constructed by relaxing the constraint, which contains the parametric optimal solution value function of the lower level problem. The upper bounding problem is obtained by testing some feasible points. Thirdly, a bounding algorithm is proposed and the convergence is also presented. Finally, to illustrate the bounding algorithm, we consider some nonlinear trilevel programming problems. It shows that the bounding algorithm proposed can terminate finitely to a ε-optimality point of the nonlinear trilevel programming problem.