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## Time Blocks Decomposition of Multistage Stochastic Optimization Problems

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Pierre Carpentier
Jean-Philippe Chancelier
Michel de Lara
Tristan Rigaut
• Function : Author
• PersonId : 1030095

#### Abstract

Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties (scenarios). Their large scale nature makes decomposition methods appealing. The most common approaches are time decomposition --- and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control --- and scenario decomposition --- like progressive hedging in stochastic programming. We present a method to decompose multistage stochastic optimization problems by time blocks, which covers both stochastic programming and stochastic dynamic programming. Once established a dynamic programming equation with value functions defined on the history space (a history is a sequence of uncertainties and controls), we provide conditions to reduce the history using a compressed state'' variable. This reduction is done by time blocks, that is, at stages that are not necessarily all the original unit stages, and we prove a reduced dynamic programming equation. Then, we apply the reduction method by time blocks to \emph{two time-scales} stochastic optimization problems and to a novel class of so-called \emph{decision-hazard-decision} problems, arising in many practical situations, like in stock management. The \emph{time blocks decomposition} scheme is as follows: we use dynamic programming at slow time scale where the slow time scale noises are supposed to be stagewise independent, and we produce slow time scale Bellman functions; then, we use stochastic programming at short time scale, within two consecutive slow time steps, with the final short time scale cost given by the slow time scale Bellman functions, and without assuming stagewise independence for the short time scale noises.

### Dates and versions

hal-01757113 , version 1 (04-04-2018)
hal-01757113 , version 2 (26-09-2018)
hal-01757113 , version 3 (11-05-2022)
hal-01757113 , version 4 (22-12-2022)

### Identifiers

• HAL Id : hal-01757113 , version 2
• ARXIV :

### Cite

Pierre Carpentier, Jean-Philippe Chancelier, Michel de Lara, Tristan Rigaut. Time Blocks Decomposition of Multistage Stochastic Optimization Problems. 2018. ⟨hal-01757113v2⟩

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