Time Blocks Decomposition of Multistage Stochastic Optimization Problems

Pierre Carpentier (1) , Jean-Philippe Chancelier (2) , Michel de Lara (2) , Tristan Rigaut (2)

1 UMA - Unité de Mathématiques Appliquées
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties. Their large scale nature makes decomposition methods appealing. We provide a method to decompose multistage stochastic optimization problems by time blocks. Our framework covers both stochastic programming and stochastic dynamic programming. We formulate multistage stochastic optimization problems over a so-called history space, with solutions being history feedbacks. We prove a general dynamic programming equation, with value functions defined on the history space. Then, we consider the question of reducing the history using a compressed " state " variable. This reduction can be done by time blocks, that is, at stages that are not necessarily all the original unit stages. We prove a reduced dynamic programming equation. Then, we apply the reduction method by time blocks to several classes of optimization problems, especially two timescales stochastic optimization problems and a novel class consisting of decision hazard decision models. Finally, we consider the case of optimization with noise process.

Mots clés: decision hazard decision; time blocks; dynamic programming; decomposition; multistage stochastic optimization; UMA; 90C39; 93E20; two time-scales; ENSTA ParisTech; MSC: 90C06; Université Paris-Saclay

Domaines: Optimisation et contrôle [math.OC]

Pierre Carpentier, Jean-Philippe Chancelier, Michel de Lara, Tristan Rigaut. Time Blocks Decomposition of Multistage Stochastic Optimization Problems. 2018. ⟨hal-01757113v1⟩