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<p>In multistage decision problems, it is often the case that an initial strategic decision (such as system design) is followed by a sequence of operational ones (operating the system). Such initial strategic decision can be seen as a parameter affecting a multistage decision problem. More generally, we study in this paper a standard multistage stochastic optimization problem depending on a parameter chosen at the initial stage. When the parameter is fixed, Stochastic Dynamic Programming provides a way to compute the optimal value of the problem. Thus, the value function depends both on the state (as usual) and on the parameter. Our aim is to investigate on the possibility to efficiently compute gradients of the value function with respect to the parameter, when these objects exist. When nondifferentiable, we propose a regularization method based on the Moreau-Yosida envelope. We present a numerical test case from day-ahead power scheduling.</p>

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Differentiability and Regularization of Parametric Convex Value Functions in Stochastic Multistage Optimization

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Adrien Le Franc,
  • Pierre Carpentier,
  • Jean-Philippe Chancelier,
  • Michel De Lara

摘要

In multistage decision problems, it is often the case that an initial strategic decision (such as system design) is followed by a sequence of operational ones (operating the system). Such initial strategic decision can be seen as a parameter affecting a multistage decision problem. More generally, we study in this paper a standard multistage stochastic optimization problem depending on a parameter chosen at the initial stage. When the parameter is fixed, Stochastic Dynamic Programming provides a way to compute the optimal value of the problem. Thus, the value function depends both on the state (as usual) and on the parameter. Our aim is to investigate on the possibility to efficiently compute gradients of the value function with respect to the parameter, when these objects exist. When nondifferentiable, we propose a regularization method based on the Moreau-Yosida envelope. We present a numerical test case from day-ahead power scheduling.