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

TL;DR

This paper explores the differentiability of parametric value functions in multistage stochastic optimization, proposing regularization techniques to compute gradients efficiently, with applications demonstrated in power scheduling.

Contribution

It introduces a regularization approach for non-differentiable value functions and analyzes their differentiability properties in multistage stochastic problems.

Findings

Regularization via Moreau-Yosida envelope enables gradient computation.

Value functions are differentiable under certain regularization conditions.

Numerical tests demonstrate practical effectiveness in power scheduling.

Abstract

In multistage decision problems, it is often the case that an initial strategic decision (such as investment) is followed by many operational ones (operating the investment). 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. 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.