Augmented lagrangian methods for solving optimization problems with stochastic-order constraints

We investigate risk-averse stochastic optimization problems with a risk-shaping constraint in the form of a stochastic-order relation. Both univariate and multivariate orders are considered. We extend the notion of a linear multivariate order, adding flexibility with respect to the controlled portion of the distributions. We propose several methods for the numerical solution of these problems based on augmented Lagrangian framework and analyze their convergence. The methods construct finite-dimensional risk-neutral approximations of the optimization problem whose solutions converge to the solution of the original problem. In the case of univariate stochastic dominance, we explore augmented Lagrangian functionals based on inverse formulations of the stochastic-order constraint. The performance of the methods is compared to other extant numerical methods and shows the numerical advantage of the augmented Lagrangian framework. The proposed numerical approach is particularly successful when applied to problems with multivariate stochastic dominance constraints.