An overview of flexibility and generalized uncertainty in optimization

Two new powerful mathematical languages, fuzzy set theory and possibility theory, have led to two optimization types that explicitly incorporate data whose values are not real-valued nor probabilistic: 1) flexible optimization and 2) optimization under generalized uncertainty. Our aim is to make clear what these two types are, make distinctions, and show how they can be applied. Flexible optimization arises when it is necessary to relax the meaning of the mathematical relation of belonging to a set (a constraint set in the context of optimization). The mathematical language of relaxed set belonging is fuzzy set theory. Optimization under generalized uncertainty arises when it is necessary to represent parameters of a model whose values are only known partially or incompletely. A natural mathematical language for the representation of partial or incomplete information about the value of a parameter is possibility theory. Flexible optimization, as delineated here, includes much of what has been called fuzzy optimization whereas optimization under generalized uncertainty includes what has been called possibilistic optimization. We explore why flexible optimization and optimization under generalized uncertainty are distinct and important types of optimization problems. Possibility theory in the context of optimization leads to two distinct types of optimization under generalized uncertainty, single distribution and dual distribution optimization. Dual (possibility/necessity pairs) distribution optimization is new. Mathematical subject classification: 90C70, 65G40.