joint work with Louis Chen, Will Ma, Karthik Natarajan, David Simchi-Levi
In this paper, we study the class of linear and discrete optimization problems in which the objective coefficients are chosen randomly from a distribution, and the goal is to evaluate robust bounds on the expected optimal value as well as the marginal distribution of the optimal solution. The set of joint distributions is assumed to be specified up to only the marginal distributions. Though the problem is NP-Hard, we establish a primal-dual analysis that yields sufficiency conditions for poly-time solvability as well as computational procedures. Application problems are also explored.
We consider settings in which the distribution of a multivariate random variable is partly ambiguous: The ambiguity lies on the level of dependence structure, and that the marginal distributions are known. We work with the set of distributions that are both close to a given reference measure in a transportation distance and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation.
We derive the quantitative stability results, under the appropriate pseudo metric, for stochastic optimization problems with distributionally robust dominance constraints induced by full random recourse. To this end, we first establish the qualitative and quantitative stability results of the optimal value function and the optimal solution set for optimization problem with $k$-th order stochastic dominance constraints under the Hausdorff metric, which extend the present results to the locally Lipschitz continuity case.