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A Benders squared (\(B^2\)) framework for infinite-horizon stochastic linear programs

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Abstract

We propose a nested decomposition scheme for infinite-horizon stochastic linear programs. Our approach can be seen as a provably convergent extension of stochastic dual dynamic programming to the infinite-horizon setting: we explore a sequence of finite-horizon problems of increasing length until we can guarantee convergence with a given confidence level. The methodology alternates between a forward pass to explore sample paths and determine trial solutions, and a backward pass to generate a polyhedral approximation of the optimal value function by computing subgradients from the dual of the scenario subproblems. A computational study on a large set of randomly generated instances for two classes of problems shows that the proposed algorithm is able to effectively solve instances of moderate size to high precision, provided that the instance structure allows the construction of what we call constant-statepolicies with satisfactory objective function value.

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Notes

  1. To be precise, the value of the optimal policy for \(P_{T(\epsilon )}\) can be found by solving (VF-L0-0), the remaining problems (VF-Lt-j) are used to identify the optimal policy.

  2. In this proof, an iteration is an execution of a forward pass and a backward pass on a sample path; this allows us a few simplifications when adapting parts of [14] to our setting.

  3. Recall that our iterations include a full backward pass; hence they include the steps between \(k\tau \) and \((k+1)\tau -1\) in [14].

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Authors and Affiliations

  1. IBM T.J. Watson, Yorktown Heights, NY, USA

    Giacomo Nannicini

  2. Laboratoire d’Informatique de Paris Nord, Université de Paris 13; and Sorbonne Paris Cité, CNRS (UMR 7538), 93430, Villetaneuse, France

    Emiliano Traversi & Roberto Wolfler Calvo

Authors
  1. Giacomo Nannicini
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  2. Emiliano Traversi
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  3. Roberto Wolfler Calvo
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Correspondence to Emiliano Traversi.

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A computational tests with different values of \(\delta \)

A computational tests with different values of \(\delta \)

In Table 5 we show the details about the computation times of the different algorithms when \(\delta \) increases.

Table 5 Extended results concerning the average CPU times for the three algorithms on the production planning with backlog problem
Full size table

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Nannicini, G., Traversi, E. & Calvo, R.W. A Benders squared (\(B^2\)) framework for infinite-horizon stochastic linear programs. Math. Prog. Comp. 13, 645–681 (2021). https://doi.org/10.1007/s12532-020-00195-2

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  • DOI: https://doi.org/10.1007/s12532-020-00195-2

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