Constrained optimization problems under uncertainty with coherent lower previsions

Abstract

ABSTRACT. We investigate a constrained optimization problem with uncertainty about constraint parameters. Our aim is to reformulate it as a (constrained) optimization problem without uncertainty. This is done by recasting the original problem as a decision problem under uncertainty. We give results for a number of different types of uncertainty modelslinear and vacuous previsions, and possibility distributions-and for two common but different optimality criteria for such decision problems-maximinity and maximality. We compare our approach with other approaches that have appeared in the literature.

Maximin and Maximal Solutions for Linear Programming Problems with Possibilistic Uncertainty

“…We can model the uncertainty using a joint upper probability defined for every subset ℰ of × ℬ by (ℰ) := sup ( , )∈ℰ ( , )-lower probabilities follow by conjugacy and lower and upper previsions by Choquet integration [7].…”

“…These problems can then be expressed as maximize subject to ≤ , ≥ 0 with given uncertainty model for ( , ) This uncertainty means that, for a given choice of , it may be uncertain whether satisfies the constraints or not, and therefore it is not clear what it means to 'maximize' in such a problem. Our approach is to transform the problem into a decision problem (Section 2) in which a fixed penalty is received if any of the constraints are broken [7]. Since this is a reformulation as a decision problem, it can then in principle be solved using a suitable optimality criterion for the uncertainty models being used, for instance maximizing expected utility when probability distributions are used, as we do in Section 3 to introduce the basic ideas.…”

Implementation of Maximin and Maximal Solutions for Linear Programming Problems Under Uncertainty

“…The following decision problem was introduced in earlier work [2,6]. The gain G x (a, b) for a particular optimization vector x and a particular realization (a, b) of the constraint parameters (A, B) is c T x if ax ≤ b and L if ax ≤ b. L is a penalty that must be smaller than c T x for any x in the outer feasibility space: those x for which there is a realization (a, b) of (A, B) such that ax ≤ b.…”

“…The solutions in this section were derived in an earlier paper [6], which is based on a more general treatment [2].…”

“…In earlier work [2] we reformulated this problem into a decision problem. Uncertainty about A ij and B i is modeled by crisp intervals [a ij , a ij ] and [b i , b i ], or by possibility distributions [3].…”