Generalized adaptive partition-based method for two-stage stochastic linear programs : convergence and generalization

TL;DR

This paper extends adaptive partition-based methods for two-stage stochastic linear programs, providing convergence proofs and broadening applicability to non-finite distributions, thus improving solution accuracy and generalization.

Contribution

It introduces a generalized APM framework with convergence guarantees and necessary conditions for adapted partitions in complex stochastic settings.

Findings

Extended APM to almost arbitrary 2SLP models

Provided necessary and sufficient conditions for adapted partitions

Proved convergence of the generalized APM method

Abstract

Adaptive Partition-based Methods (APM) are numerical methods to solve two-stage stochastic linear problems (2SLP). The core idea is to iteratively construct an adapted partition of the space of alea in order to aggregate scenarios while conserving the true value of the cost-to-go for the current first-stage control. Relying on the normal fan of the dual admissible set, we extend the classical and generalized APM method by i) extending the method to almost arbitrary 2SLP, ii) giving a necessary and sufficient condition for a partition to be adapted even for non-finite distribution, and iii) proving the convergence of the method. We give some additional insights by linking APM to the L-shaped algorithm.