Optimization under uncertainty

Abstract: One of the most ubiquitous problems in optimization is that of finding all the elements of a finite set at which a function $f$ attains its minimum (or maximum) on that set. When the codomain of $f$ is equipped with a reflexive, anti-symmetric and transitive relation, it is easy to specify, implement and verify generic solutions for this problem. But what if $f$ is affected by uncertainties? What if one seeks values that minimize more than one $f$ or if $f$ does not return a single result but a set of ``possible results'' or perhaps a probability distribution on possible results? This situation is very common in integrated assessment and optimal design and developing trustable solution methods for optimization under uncertainty requires one to formulate the above questions rigorously. We show how functional programming can help formulating such questions and apply it to specify and test solution methods for the case in which optimization is affected by two conceptually different kinds of uncertainty: \it{value} and \it{functorial} uncertainty.