Computation of Efficient Solutions of Stochastic Optimization Problems with Applications to Regression and Scenario Analysis

Abstract

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 $$ minnimize F\left( x \right) = Eu\left( {A\left( \omega \right)x - b\left( \omega \right)} \right) s.t. x \in D $$ (1) where u is a convex loss function on IRm, (A(ω),b(ω)) is a random (m,n+1)-matrix, “E” denotes the expectation operator and D is a convex subset of IRn. 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.