Computation of efficient solutions of discretely distributed stochastic optimization problems

Abstract

In engineering and economics often a certain vectorx of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the outputA(ω) x of a stochastic linear systemx→A(ω)x and a desired stochastic target vectorb(ω) are minimal. Hence, one has the following stochastic linear optimization problem minimizeF(x)=Eu(A(ω)x b(ω)) s.t.xεD, (1) whereu is a convex loss function on ℝ^m, (A(ω), b(ω)) is a random (m,n + 1)-matrix, “E” denotes the expectation operator andD is a convex subset of ℝⁿ. Concrete problems of this type are e.g. stochastic linear programs with recourse, error minimization and optimal design problems, acid rain abatement methods, problems in scenario analysis and non-least square regression analysis.

Solving (1), the loss functionu should be exactly known. However, in practice mostly there is some uncertainty in assigning appropriate penalty costs to the deviation between the outputA (ω)x and the targetb(ω). For finding in this situation solutions “hedging against uncertainty” a set of so-called efficient points of (1) is defined and a numerical procedure for determining these compromise solutions is derived. Several applications are discussed.