A framework of distributionally robust possibilistic optimization

In this paper, an optimization problem with uncertain constraint coefficients is considered. Possibility theory is used to model the uncertainty. Namely, a joint possibility distribution in constraint coefficient realizations, called scenarios, is specified. This possibility distribution induces a necessity measure in scenario set, which in turn describes an ambiguity set of probability distributions in scenario set. The distributionally robust approach is then used to convert the imprecise constraints into deterministic equivalents. Namely, the left-hand side of an imprecise constraint is evaluated by using a risk measure with respect to the worst probability distribution that can occur. In this paper, the Conditional Value at Risk is used as the risk measure, which generalizes the strict robust and expected value approaches, commonly used in literature. A general framework for solving such a class of problems is described. Some cases which can be solved in polynomial time are identified.