Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs

TL;DR

This paper analyzes sampling-based decomposition methods for risk-averse multistage stochastic convex programs, proving convergence under various conditions and providing formulas for efficient implementation of stochastic dual dynamic programming.

Contribution

It introduces a formula for computing cuts in decomposition methods and proves their almost sure convergence for both convex and linear risk-averse multistage stochastic programs.

Findings

Formula for cut computation in decomposition methods

Almost sure convergence under relatively complete recourse

Extension to interstage dependent stochastic processes

Abstract

We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. We prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds. We also prove the almost sure convergence of these algorithms when applied to risk-averse multistage stochastic linear programs that do not satisfy the relatively complete recourse assumption. The analysis is first done assuming the underlying stochastic process is interstage independent and discrete, with a finite set of possible realizations at each stage. We then indicate two ways of extending the methods and convergence analysis to the case when the process is interstage dependent.