Scenario Reduction Techniques in Stochastic Programming WernerR¨omisch Humboldt-University Berlin, Department of Mathematics, Rudower Chaussee 25, 12489 Berlin, Germany Abstract. Stochastic programming problems appear as mathematical models for optimization problems under stochastic uncertainty. Most computational approaches for solving such models are based on approx- imating the underlying probability distribution by a probability mea- sure with finite support. Since the computational complexity for solving stochastic programs gets worse when increasing the number of atoms (or scenarios), it is sometimes necessary to reduce their number. Techniques for scenario reduction often require fast heuristics for solving combinato- rial subproblems. Available techniques are reviewed and open problems are discussed. 1 Introduction Many stochastic programming problems may be reformulated in the form min  E(f 0 (x,ξ )) =  R s f 0 (x,ξ )P (dξ ): x ∈ X  , (1) where X denotes a closed subset of R m , the function f 0 maps from R m × R s to the extended real numbers R = R ∪ {−∞, +∞}, E denotes expectation with respect to P and P is a probability distribution on R s . For example, models of the form (1) appear as follows in economic applica- tions. Let ξ = {ξ t } T t=1 denote a discrete-time stochastic process of d-dimensional random vectors ξ t at each t ∈{1,...,T } and assume that decisions x t have to be made such that the total costs appearing in an economic process are minimal. Such an optimization model may often be formulated as min  E  T  t=1 f t (x t ,ξ t )  : x t ∈ X t , t−1  τ =0 A tτ (ξ t )x t−τ = h t (ξ t ),t =1,...,T  . (2) Typically, the sets X t are polyhedral, but they may also contain integrality con- ditions. In addition, the decision vector (x 1 ,...,x T ) has to satisfy a dynamic constraint (i.e. x t depends on the former decisions) and certain balancing condi- tions. The matrices A tτ (ξ t ), τ =0,...,t − 1, (e.g. containing technical parame- ters) and the right-hand sides h t (ξ t ) (e.g. demands) are (partially) random. The functions f t describe the costs at time t and may also be (partially) random (e.g.