Two‐stage stochastic optimization problems with stochastic ordering constraints on the recourse

We consider two‐stage risk‐averse stochastic optimization problems with a stochastic ordering constraint on the recourse function. Two new characterizations of the increasing convex order relation are provided. They are based on conditional expectations and on integrated quantile functions: a counterpart of the Lorenz function. We propose two decomposition methods to solve the problems and prove their convergence. Our methods exploit the decomposition structure of the risk‐neutral two‐stage problems and construct successive approximations of the stochastic ordering constraints. Numerical results confirm the efficiency of the methods.