In engineering and economics often a certain vector x of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the output a(ω)x of a stochastic linear system x⇒A(ω)x and a desired stochastic target vector b(ω) are minimal. Hence, one has the following stochastic linear optimization problem: minimize F(x)=Eu(A(ω)x-b(ω)) s.t. x∈D, where u is a convex loss function on ℝm, (A(ω),b(ω)) is a random (m,n+1)-matrix, ‘E’ denotes the expectation operator and D 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 the above equation, the loss function u should be exactly known. However, in practice mostly there is some uncertainty in assigning appropriate penalty costs to the deviation between the output A(ω)x and the target b(ω). For finding in this situation solutions ‘hedging against uncertainty’ a set of so-called efficient points of the equation is defined and a numerical procedure for determining these compromise solutions is derived. Several applications are discussed.
