On the Convergence of L-shaped Algorithms for Two-Stage Stochastic Programming

TL;DR

This paper develops and analyzes a variant of the L-shaped algorithm for two-stage stochastic programming, accommodating inexact objective and gradient estimates, with proven convergence properties and empirical validation.

Contribution

It introduces a new inexact L-shaped algorithm with convergence analysis and sample complexity results for stochastic programming problems.

Findings

Algorithm converges to a neighborhood of optimality.

Provides bounds on iterations needed for approximate solutions.

Numerical results show superior performance over classic solvers.

Abstract

In this paper, we design, analyze, and implement a variant of the two-loop L-shaped algorithms for solving two-stage stochastic programming problems that arise from important application areas including revenue management and power systems. We consider the setting in which it is intractable to compute exact objective function and (sub)gradient information, and instead, only estimates of objective function and (sub)gradient values are available. Under common assumptions including fixed recourse and bounded (sub)gradients, the algorithm generates a sequence of iterates that converge to a neighborhood of optimality, where the radius of the convergence neighborhood depends on the level of the inexactness of objective function estimates. The number of outer and inner iterations needed to find an approximate optimal iterate is provided. Finally, we show a sample complexity result for the algorithm with a Polyak-type step-size policy that can be extended to analyze other situations. We also present a numerical study that verifies our theoretical results and demonstrates the superior empirical performance of our proposed algorithms over classic solvers.