An l¦1 penalty function approach to the nonlinear bilevel programming problem

Abstract

The nonlinear bilevel programming problem is a constrained optimization problem defined over two vector of unknowns, x and y. Feasibility constraints on (x,y) include the requirement that y is a solution of another optimization problem, called the inner problem, which is parameterized by x. The bilevel problem is very difficult to solve, and few algorithms have been published for the nonlinear problem. Therefore, instead of solving the bilevel problem directly, a "simpler", related problem is solved . This problem is defined by replacing the solution constraint in the bilevel problem with a set of conditions which must be satisfied at a minimum point of the inner problem. The resulting one level mathematical program is solved using an exact penalty function technique, which involves finding solutions to a series of unconstrained problems. These problems are usually nonconvex and nondifferentiable. Each problem is solved within a trust region framework, and specialized techniques are developed to overcome difficulties due to the nondifferentiabilities. A unique approach is developed to resolve degeneracy in the penalty function problems. The algorithm is proven to converge to a minimum point of the penalty function. Testing results are presented and analyzed.