Two-Stage Optimization Problems with Multivariate Stochastic Order Constraints

We propose a two-stage risk-averse stochastic optimization problem with a stochastic-order constraint on a vector-valued function of the second-stage decisions. This model is motivated by a multiobjective second-stage problem. We formulate optimality conditions for the problem and analyse the Lagrangian relaxation of the order constraint. We propose two decomposition methods to solve the problems and prove their convergence. The methods are based on Lagrangian relaxation of the order constraints and on a construction of successive risk-neutral two-stage problems. Additionally, we propose a new combinatorial method for verification of the multivariate order relation, which is a key part of both methods. We analyse a supply chain problem using our model and we apply our methods to solve the optimization problem. Numerical results confirm the efficiency of the proposed methods.