REGULARIZED MATHEMATICAL PROGRAMS WITH STOCHASTIC EQUILIBRIUM CONSTRAINTS: ESTIMATING STRUCTURAL DEMAND MODELS

The article considers a particular class of optimization problems involving set-valued stochastic equilibrium constraints. We develop a solution procedure that relies on an approximation scheme for the equilibrium constraints. Based on regularization, we replace the approximated equilibrium constraints by those involving only single-valued Lipschitz continuous functions. In addition, sampling has the further effect of replacing the "simplified" equilibrium constraints by more manageable ones obtained by implicitly discretizing the (given) probability measure so as to render the problem computationally tractable. Convergence is obtained by relying, in particular, on the graphical convergence of the approximated equilibrium constraints. The problem of estimating the characteristics of a demand model, a widely studied problem in microeconometrics, serves both as motivation and illustration of the regularization and sampling procedure.