Distributionally robust possibilistic optimization problems

TL;DR

This paper introduces a distributionally robust possibilistic optimization framework that handles uncertain linear constraints with imprecise probability information, transforming them into tractable deterministic problems.

Contribution

It develops a novel approach combining possibility theory with distributional robustness to model and solve uncertain optimization problems efficiently.

Findings

The approach yields linear or second order cone reformulations.

It is computationally tractable for linear programming.

The method effectively manages imprecise probabilistic information.

Abstract

In this paper a class of optimization problems with uncertain linear constraints is discussed. It is assumed that the constraint coefficients are random vectors whose probability distributions are only partially known. Possibility theory is used to model the imprecise probabilities. In one of the interpretations, a possibility distribution (a membership function of a fuzzy set) in the set of coefficient realizations induces a necessity measure, which in turn defines a family of probability distributions in this set. The distributionally robust approach is then used to transform the imprecise constraints into deterministic counterparts. Namely, the uncertain left-had side of each constraint is replaced with the expected value with respect to the worst probability distribution that can occur. It is shown how to represent the resulting problem by using linear or second order cone constraints. This leads to problems which are computationally tractable for a wide class of optimization models, in particular for linear programming.