On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems

Abstract

In this paper, we study a new exact and smooth penalty function for the nonlinear mixed discrete programming problem by augumenting only one variable no matter how many constraints. Through the smooth and exact penalty function, we can transform the nonlinear mixed discrete programming problem into an unconstrained optimization model. We demonstrate that under mild conditions, when the penalty parameter is sufficiently large, optimizers of this penalty function are precisely the optimizers of the nonlinear mixed discrete programming problem. Alternatively, under some mild assumptions, the local exactness property is also presented. The numerical results demonstrate that the new penalty function is an effective and promising approach. As important applications, we solve an increasingly popular search engine advertising problem via the new proposed penalty function.