Interior-Point Algorithms, Penalty Methods and Equilibrium Problems

In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPEC’s)—as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties, present an example from game theory where this makes a difference in practice, and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interior-point algorithms encounter in general.