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. A solution procedure is developed by relying on an approximation scheme for the equilibrium constraints, based on regularization, that replaces them by equilibrium constraints 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 micro-economics, serves both as motivation and illustration of the regularization and sampling procedure. REGULARIZED MATHEMATICAL PROGRAMS WITH STOCHASTIC EQUILIBRIUM CONSTRAINTS: ESTIMATING STRUCTURAL DEMAND MODELS XIAOJUN CHEN∗, HAILIN SUN† , AND ROGER J-B WETS‡