Optimization with multivariate stochastic dominance constraints (Q959965)

Considered is the following optimization model with stochastic dominance constraints: (1) \(\max \mathbb{E}\left[ H\left( z\right) \right]\) subject to (2) \(G\left( z\right) \succsim _{\left( 2\right) }Y,\) (3) \(z\in Z_{0}\) where \(Z_{0}\) is a convex closed subset of a Banach space \(\mathcal{L}\), and \(G\) and \(H\) are continuous operations from \(\mathcal{L}\) to the space of integrable random variables \(\mathcal{L}_{1}\left( \Omega ,\mathcal{F},P,R\right) \). The authors investigate the case when \(G\left( z\right) \) and \(Y\) in (2) are \(m\)-dimensional random vectors, rather than scalar variables. In Section 2 they introduce a multivariate stochastic dominance constraint in a form of a linear stochastic order and construct a generator of this order. In Section 3 the authors analyze problem (1)-(3) with a multivariate constraint (2), under convexity assumptions and develop necessary and sufficient conditions of optimality. Also, they define the dual problem and prove the duality theorem. In Section 4 the problem (1)-(3) is analyzed in the case when the mappings \(H\) and \(G\) are not concave, but differentiable.