Numerical evaluation of approximation methods in stochastic programming

INFORMS Journal on Computing

2013

I n the context of multistage stochastic optimization problems, we propose a hybrid strategy for generalizing to nonlinear decision rules, using machine learning, a finite data set of constrained vector-valued recourse decisions optimized using scenario-tree techniques from multistage stochastic programming. The decision rules are based on a statistical model inferred from a given scenario-tree solution and are selected by out-of-sample simulation given the true problem. Because the learned rules depend on the given scenario tree, we repeat the procedure for a large number of randomly generated scenario trees and then select the best solution (policy) found for the true problem. The scheme leads to an ex post selection of the scenario tree itself. Numerical tests evaluate the dependence of the approach on the machine learning aspects and show cases where one can obtain near-optimal solutions, starting with a "weak" scenario-tree generator that randomizes the branching structure of the trees.

“…Consider the following problem, studied by Hilli and Pennanen (2008) and Küchler and Vigerske (2010), where ρ is a risk-aversion parameter and η a budget parameter:…”

Section: Studied Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Scenario Trees and Policy Selection for Multistage Stochastic Programming Using Machine Learning

INFORMS Journal on Computing

2013

“…, ξ t−1 )) ∈ U t evaluated and implemented online on new scenarios. Repair procedures are also suggested in Küchler and Vigerske (2010) as a means of restoring the feasibility of decisions extracted from a tree and directly applied on test scenarios.…”

Section: Inference and Evaluation Of A Policy From A Scenario Treementioning

confidence: 99%

“…The section starts by the formulation of a multistage stochastic program that various researchers have presented as difficult for scenario tree methods (Hilli & Pennanen, 2008;Koivu & Pennanen, 2010;Küchler & Vigerske, 2010). Several instances of the problem will be addressed, including instances on horizons considered as almost unmanageable by scenario tree methods.…”

Section: Case Studymentioning

confidence: 99%

Multistage Stochastic Programming

Decision Theory Models for Applications in Artificial Intelligence

2012

In this chapter, we present the multistage stochastic programming framework for sequential decision making under uncertainty. We discuss its differences with Markov Decision Processes, from the point of view of decision models and solution algorithms. We describe the standard technique for solving approximately multistage stochastic problems, based on a discretization of the disturbance space called scenario tree. We insist on a critical issue of the approach; the decisions can be very sensitive to the parameters of the scenario tree, whereas no efficient tool for checking the quality of approximate solutions exists. In this chapter, we show how supervised learning techniques can be used to evaluate reliably the quality of an approximation, and thus facilitate the selection of a good scenario tree. The framework and solution techniques presented in the chapter are explained and detailed on several examples. Along the way, we define notions from decision theory that can be used to quantify, for a particular problem, the advantage of switching to a more sophisticated decision model.

“…The need for generalizing decisions on a subset of scenarios to an admissible policy is well recognized in the stochastic programming literature [11]. Some authors addressing this question propose to assign to a new scenario the decisions associated to the nearest scenario of the approximate solution [20,21], thus essentially reducing the generalization problem to the one of defining a priori a measurable similarity metric in the scenario space (we note that [21] also proposes several variants for a projection step restoring the feasibility of the decisions). Put in perspective of the present framework, this amounts to adopt, without model selection, the nearest neighbor approach to regression [22] -arguably one of the most unstable prediction algorithm [17].…”

Section: Related Workmentioning

confidence: 99%

Bounds for Multistage Stochastic Programs Using Supervised Learning Strategies

Stochastic Algorithms: Foundations and Applications

2009

We propose a generic method for obtaining quickly good upper bounds on the minimal value of a multistage stochastic program. The method is based on the simulation of a feasible decision policy, synthesized by a strategy relying on any scenario tree approximation from stochastic programming and on supervised learning techniques from machine learning. 1 Context Let Ω denote a measurable space equipped with a sigma algebra B of subsets of Ω, defined as follows. For t = 0, 1,. .. , T − 1, we let ξ t be a random variable valued in a subset of an Euclidian space Ξ t , and let B t denote the sigma algebra generated by the collection of random variables ξ [0: t−1] def = {ξ 0 ,. .. , ξ t−1 }, with B 0 = {∅, Ω} corresponding to the trivial sigma algebra. Then we set B T −1 = B. Note that B 0 ⊂ B 1 ⊂ • • • ⊂ B T −1 form a filtration; without loss of generality, we can assume that the inclusions are proper-that is, ξ t cannot be reduced to a function of ξ [0: t−1]. Let π t : Ξ → U t denote a B t-measurable mapping from the product space Ξ = Ξ 0 × • • • × Ξ T −1 to an Euclidian space U t , and let Π t denote the class of such mappings. Of course Π 0 is a class of real-valued constant functions. We equip the measurable space (Ω, B) with the probability measure P and consider the following optimization program, which is a multistage stochastic program put in abstract form: S : min π∈Π E {f (ξ, π(ξ))} subject to π t (ξ) ∈ U t (ξ) almost surely. (1) Here ξ denotes the random vector [ξ 0. .. ξ T −1 ] valued in Ξ, and π is the mapping from Ξ to the product space U = U 0 × • • • × U T −1 defined by π(ξ) = [π 0 (ξ). .. π T −1 (ξ)] with π t ∈ Π t. We call such π an implementable policy and let Π denote the class of implementable policies. The function f : Ξ × U → R ∪ {±∞} is B-measurable and such that f (•, ξ) is convex for each ξ. It can be interpreted as a multi-period cost function. The sets U t (ξ) are defined as B t-measurable closed convex subsets of U t. A set U t (ξ) may implicitly depend on π 0 (ξ),. .. , π t−1 (ξ) viewed as B t-measurable