Inexact cuts in SDDP applied to multistage stochastic nondifferentiable problems

TL;DR

This paper extends the Inexact Stochastic Dual Dynamic Programming (ISDDP) method to handle multistage stochastic problems with nondifferentiable convex functions, providing convergence analysis and practical formulas for inexact cuts.

Contribution

It introduces inexact cuts for nondifferentiable convex value functions and adapts ISDDP to these problems, with proven convergence properties.

Findings

ISDDP converges to an optimal policy with vanishing errors.

Upper and lower bounds on the optimal value are within 3*epsilon*T for bounded errors.

The paper provides formulas for inexact cuts in nondifferentiable convex optimization.

Abstract

In [13], an Inexact variant of Stochastic Dual Dynamic Programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the method. That variant of SDDP was studied in [13] for linear and for differentiable nonlinear Multistage Stochastic Programs (MSPs). In this paper, we extend ISDDP to nondifferentiable MSPs. We first provide formulas for inexact cuts for value functions of convex nondifferentiable optimization problems. We then combine these cuts with SDDP to describe ISDDP for nondifferentiable MSPs and analyze the convergence of the method. More precisely, for a problem with T stages, we show that for errors bounded from above by epsilon, the limit superior and limit inferior of sequences of upper and lower bounds on the optimal value of the problem are at most at distance 3*epsilon*T to the optimal value and that for asymptotically vanishing errors ISDDP converges to an optimal policy. [13] V. Guigues, Inexact cuts in Stochastic Dual Dynamic Programming, Siam Journal on Optimization, 30(1), 407-438, 2020.