Interval Analysis, Fuzzy Set Theory and Possibility Theory in Optimization

This study explores the interconnections between interval analysis, fuzzy interval analysis, and interval, fuzzy, and possibility optimization. The key ideas considered are: (1) Fuzzy and possibility optimization are important methods in applied optimization problems, (2) Fuzzy sets and possibility distributions are distinct which in the context of optimization lead to ‡exible optimization on one hand and optimization under uncertainty on the other, (3) There are two ways to view possibility analysis leading to two distinct types of possibility optimization under uncertainty, single and dual distribution optimization, and (4) The semantic (meaning) of constraint set in the presence of fuzziness and possibility needs to be known a-priori in order to determine what the constraint set is and how to compute its elements. The thesis of this exposition is that optimization problems that intend to model real problems are very (most) often satis…cing and epistemic and that the mathematical language of fuzzy sets and possibility theory are well-suited theories for stating and solving a wide classes of optimization problems.