An Objective Penalty Method for Optimistic Bilevel Programming Problems

In this paper,we consider an optimistic nonlinear bilevel programming problem.Under some conditions,we first show that the sequence of solutions to penalty problems converges to the optimal solution of the original bilevel programming problem.We then present an objective penalty method to solve such a problem.Finally,some numerical experiments are performed to illustrate its feasibility.

A New Objective Penalty Function Approach for Solving Constrained Minimax Problems

In this paper,a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints.This new objective penalty function combines the objective penalty and constraint penalty.By the new objective penalty function,a constrained minimax problem is converted to minimizations of a sequence of continuously differentiable functions with a simple box constraint.One can thus apply any efficient gradient minimization methods to solve the minimizations with box constraint at each step of the sequence.Some relationships between the original constrained minimax problem and the corresponding minimization problems with box constraint are established.Based on these results,an algorithm for finding a global solution of the constrained minimax problems is proposed by integrating the particular structure of minimax problems and its global convergence is proved under some conditions.Furthermore,an algorithm is developed for finding a local solution of the constrained minimax problems,with its convergence proved under certain conditions.Preliminary results of numerical experiments with well-known test problems show that satisfactorilyapproximate solutions for some constrained minimax problems can be obtained.