Inexact cuts in Deterministic and Stochastic Dual Dynamic Programming applied to linear optimization problems

TL;DR

This paper extends Dual Dynamic Programming to handle inexact solutions in linear and stochastic settings, providing convergence guarantees and demonstrating faster policy computation in numerical experiments.

Contribution

It introduces IDDP-LP and ISDDP-LP, inexact variants with convergence proofs, for linear and stochastic linear dynamic programming problems.

Findings

ISDDP-LP converges with bounded and vanishing errors.

Numerical results show faster policy computation with ISDDP-LP.

Inexact methods maintain solution quality while reducing computation time.

Abstract

We introduce an extension of Dual Dynamic Programming (DDP) to solve linear dynamic programming equations. We call this extension IDDP-LP which applies to situations where some or all primal and dual subproblems to be solved along the iterations of the method are solved with a bounded error (inexactly). We provide convergence theorems both in the case when errors are bounded and for asymptotically vanishing errors. We extend the analysis to stochastic linear dynamic programming equations, introducing Inexact Stochastic Dual Dynamic Programming for linear programs (ISDDP-LP), an inexact variant of SDDP applied to linear programs corresponding to the situation where some or all problems to be solved in the forward and backward passes of SDDP are solved approximately. We also provide convergence theorems for ISDDP-LP for bounded and asymptotically vanishing errors. Finally, we present the results of numerical experiments comparing SDDP and ISSDP-LP on a portfolio problem with direct transation costs modelled as a multistage stochastic linear optimization problem. On these experiments, ISDDP-LP allows us to obtain a good policy faster than SDDP.