The Scenario Generation Algorithm for Multistage Stochastic Linear Programming

Abstract

A multistage stochastic linear program (MSLP) is a model of sequential stochastic optimization where the objective and constraints are linear. When any of the random variables used in the MSLP are continuous, the problem is infinite dimensional. In order to numerically tackle such a problem we usually replace it with a finite dimensional approximation. Even when all the random variables have finite support, the problem is often computationally intractable and must be approximated by a problem of smaller dimension. One of the primary challenges in the field of stochastic programming deals with discovering effective ways to evaluate the importance of scenarios, and to use that information to trim the scenario tree in such a way that the solution to the smaller optimization problem is not much different than the problem stated with the original tree. The Scenario Generation (SG) algorithm proposed in this paper is a finite element method that addresses this problem for the class of MSLP with random right-hand sides.

Stability and Scenario Trees for Multistage Stochastic Programs

“…Theorem 2.1 Let (A1) -(A5) be satisfied for some pair (r, r ′ ) satisfying (2). Then the solution set S(ξ) of (3) is nonempty, convex and compact with respect to the weak topology…”

“…They are based on several different principles. We mention here (i) bound-based constructions [2,10,15,29], (ii) Monte Carlo-based schemes [3,41,43] or Quasi Monte Carlo-based methods [32,33], (iii) (EVPI-based) Sampling within decomposition schemes [4,7,21,26], (iv) the target/moment-matching principle [24,25], and (v) probability metric based approximations [17,19,22,23,34].…”

“…The approach of (i) relies on constructing discrete probability measures that correspond to lower and upper bounds (under certain assumptions on the model and the stochastic input) and on refinement strategies. The recent paper/monograph [2,29] belonging to (i) also offer convergence arguments (restricted to linear models containing only stochasticity in right-hand sides in [2] and to convex models whose stochasticity is assumed to follow some linear block-diagonal autoregressive process with compact supports in [29]). The Monte Carlo-based methods in (ii) utilize conditional sampling schemes and lead to a large number of (pseudo) random number generator calls for conditional distributions.…”

Scenario Reduction Techniques in Stochastic Programming

“…, T . (2) Typically, the sets X t are polyhedral, but they may also contain integrality conditions. In addition, the decision vector (x 1 , .…”

“…due to uncertain market prices). The objective in (2) consists in minimizing the expected total costs.…”