Smoothing mathematical programmes with equilibrium constraints via neural network function

We propose a smoothing approach based on neural network function to solve a mathematical programme with equilibrium constraints (MPEC) in which the constraints are defined by a parametric variational inequality (PVI). We reformulate MPEC as an equivalent one level non-smooth optimisation problem. Then, this non-smooth optimisation problem will transfer to a sequence of smooth optimisation problems that can be solved by standard available software for constrained optimisation. Our results obtained in this paper continue to hold for any mathematical programme with parametric nonlinear complementarity/mixed complementarity constraints. Also, we test the performance of the proposed smoothing approach on a set of well-known problems and give some comparisons between our approach and other smoothing approaches. We are hoping our smoothing approach via neural network function will provide a basis for future applications work in this area, and generate some dynamic interactions between algorithmic developers and modellers/practitioners in the MPEC field.