De Cooman, Gert

Quaeghebeur, Erik

Shariatmadar, Keivan

English

de Cooman, Gert

Ghent University Academic Bibliography

6th International Symposium on Imprecise Probability: Theories and Applications, Durham, United Kingdom, 2009A constrained optimization problem under uncertaintyKeivan Shariatmadar, Erik Quaeghebeur & Gert de CoomanGhent University, SYSTeMS Research Group{Keivan.Shariatmadar,Erik.Quaeghebeur,Gert.deCooman}@UGent.beAbstractConsider the following optimization problem: maximize a bounded real-valued function f – defined on a setX – overall x inX that satisfy the constraint xRY , where Y is a random variable taking values in a set Y and R is a relation onX ×Y . The aim of our active research is to reduce this problem to a (constrained) optimization problem from which theuncertainties present in the description of the constraint are eliminated.We investigate what results can be obtained for different types of uncertainty models for the random variable Y – linearprevisions, vacuous previsions, possibility distributions, p-boxes, etc. [see, e.g., 1, 2, 4] – and for two different optimalitycriteria – maximinity and maximality [see, e.g., 3]. We work with generalX , Y , and R for the combinations of uncertaintymodel and optimality criterion that allow it and restrict our attention to more concrete situations otherwise.In our poster, we will present the problem statement, give illustrated solutions for the most interesting cases we haveinvestigated, as well as discuss the strengths and weaknesses of our approach. In the remainder of this abstract, we sketchthis approach and give the solutions of two of the cases we have already fully investigated.The first thing we do is reformulate the optimization problem as a decision problem. To wit, with every x inX we associatea utility function Gx on Y that gives the constant value f (x) for y in Y such that xRy and that gives a penalty-value Lotherwise (i.e., when x 6Ry). As the name suggest, it penalizes the fact that x such that x 6Ry are considered; we thereforechoose L< inf f . The uncertainty model associated to Y is formulated as a coherent lower prevision P (and its conjugateupper prevision P) for a sufficiently rich set of gambles on Y [see, e.g., 4, for terminology]. Using the maximinity criterion,the optimal values forX are those that maximize P(Gx). Using the more conservative maximality criterion, the optimalvalues forX are those for which minz∈X P(Gx−Gz)≥ 0. Both finding the prevision on the sufficiently rich set of gambles{Gx : x ∈X } or {Gx−Gz : x,z ∈X } and checking the optimality criteria are, in general, nontrivial steps.For the general case, when Y is described by a linear prevision P, we find that under both maximinity and maximality theoptimal x are those that maximize the function onX that takes the value ( f (x)−L)P({y ∈ Y : xRy}) in x.For the concrete caseX = Y := R and R :=≤, and when Y is described by a triangular possibility distribution with basis[a,b]⊂ R and top c ∈ (a,b), we find that under maximinity the optimal x are those that maximize the function on (−∞,c]that takes the value f (x) for x≤ a and x−ac−aL+ c−xc−a f (x) for x in (a,c].Keywords. Constrained optimization, maximinity, maximality, linear prevision, vacuous prevision, possibility distribution,p-box.References[1] Didier Dubois and Henri Prade. Théorie des possibilités. Masson, 2nd edition, 1988.[2] Scott Ferson, Vladik Kreinovich, Lev Ginzburg, Davis S. Myers, and Kari Sentz. Constructing probability boxes andDempster-Shafer structures. Technical Report SAND2002-4015, Sandia National Laboratories, 2002.[3] Matthias C. M. Troffaes. Decision making under uncertainty using imprecise probabilities. International Journal ofApproximate Reasoning, 45(1):17–29, 5 2007. doi:10.1016/j.ijar.2006.06.001.[4] Peter Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.

A constrained optimization problem under uncertainty

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A constrained optimization problem under uncertainty

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