Complexity of stochastic dual dynamic programming

Stochastic dual dynamic programming is a cutting plane type algorithm for multi-stage stochastic optimization originated about 30 years ago. In spite of its popularity in practice, there does not exist any analysis on the convergence rates of this method. In this paper, we first establish the number...

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Bibliographic Details
Published in:Mathematical programming Vol. 191; no. 2; pp. 717 - 754
Main Author: Lan, Guanghui
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2022
Springer
Springer Nature B.V
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Complexity
Decision making
Dynamic programming
Flying-machines
Full Length Paper
Iterative methods
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimal control
Optimization
Stochastic processes
Theoretical
90C06
93A14
90C15
49M37
90C25
90C22
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:Stochastic dual dynamic programming is a cutting plane type algorithm for multi-stage stochastic optimization originated about 30 years ago. In spite of its popularity in practice, there does not exist any analysis on the convergence rates of this method. In this paper, we first establish the number of iterations, i.e., iteration complexity, required by a basic dual dynamic programming method for solving single-scenario multi-stage optimization problems, by introducing novel mathematical tools including the saturation of search points. We then refine these basic tools and establish the iteration complexity for an explorative dual dynamic programing method proposed herein and the classic stochastic dual dynamic programming method for solving more general multi-stage stochastic optimization problems under the standard stage-wise independence assumption. Our results indicate that the complexity of these methods mildly increases with the number of stages T , in fact linearly dependent on T for discounted problems. Therefore, they are efficient for strategic decision making which involves a large number of stages, but with a relatively small number of decision variables in each stage. Without explicitly discretizing the state and action spaces, these methods might also be pertinent to the related reinforcement learning and stochastic control areas.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-020-01567-1