Toward breaking the curse of dimensionality: An fptas for stochastic dynamic programs with multidimensional actions and scalar states

Nir Halman, Giacomo Nannicini

Research output: Contribution to journalArticlepeer-review

Abstract

We propose a fully polynomial-time approximation scheme (FPTAS) for stochastic dynamic programs with multidimensional action, scalar state, convex costs, and linear state transition function. The action spaces are polyhedral and described by parametric linear programs. This type of problem finds applications in the area of optimal planning under uncertainty, and can be thought of as the problem of optimally managing a single nondiscrete resource over a finite time horizon. We show that under a value oracle model for the cost functions this result for one-dimensional state space is the "best possible," because a similar dynamic programming model with two-dimensional state space does not admit a polynomial-time approximation scheme. The FPTAS relies on the solution of polynomial-sized linear programs to recursively compute an approximation of the value function at each stage. Our paper enlarges the class of dynamic programs that admit an FPTAS by showing, under suitable conditions, how to deal with multidimensional action spaces and with vectors of continuous random variables with bounded support. These results bring us one step closer to overcoming the curse of dimensionality of dynamic programming.

Original languageEnglish
Pages (from-to)1131-1163
Number of pages33
JournalSIAM Journal on Optimization
Volume29
Issue number2
DOIs
StatePublished - 2019
Externally publishedYes

Keywords

  • Approximation algorithms
  • Dynamic programming
  • Fully polynomial-time approximation scheme
  • Multistage linear programming

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Applied Mathematics

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