Solving stochastic optimization problems with stochastic constraints: an application in network design

Recently sample-path methods have been successfully used in solving challenging simulation optimization and stochastic equilibrium problems. We deal with a variant of these methods to solve stochastic optimization problems with stochastic constraints. Using optimality conditions, we convert the problem to a stochastic variational inequality. We outline a set of sufficient conditions for the almost-sure convergence of the method. We also illustrate an application by using the method to solve a network design problem. We find optimal arc capacities for a stochastic network (in which the demand and supply at each node is random) that minimize the sum of the capacity allocation cost and a measure of the expected shortfall in capacity.