Probabilistic analysis of dual decomposition on two-stage stochastic integer programs

TL;DR

This paper provides the first average-case theoretical analysis of Branch-and-Price for two-stage stochastic integer programs, showing quasi-polynomial bounds and a shrinking integrality gap under a stochastic-input model.

Contribution

It introduces an average-case analysis framework for Branch-and-Price, demonstrating improved theoretical bounds and integrality gap behavior in typical instances.

Findings

Branch-and-Price explores at most n^O(log s) nodes with high probability.

The integrality gap of the LP relaxation shrinks at rate O((logs log^2 n)/n).

The integrality gap grows only logarithmically with the number of scenarios on average.

Abstract

Two-stage stochastic integer programs provide a powerful framework for modeling decision-making under uncertainty, but they are notoriously difficult to solve at scale due to their high dimensionality and intrinsic nonconvexity. Decomposition-based algorithms such as Benders methods and Branch-and-Price (related dual decomposition methods) have become standard computational approaches for such problems and demonstrate excellent empirical performance in practice. Despite their widespread use, however, existing theoretical guarantees are almost exclusively based on worst-case analyses, which predict exponential convergence behavior in the problem dimension and fail to explain the strong performance observed in practice. In this paper, we present the first average-case analysis of Branch-and-Price for a broad class of two-stage stochastic binary integer programs. We study a stochastic-input model in which objective coefficients and constraint matrices are drawn at random and right-hand-side vectors scale with the decision dimension, while the number of constraints per scenario is fixed. Under this model, we prove that, with high probability, Branch-and-Price explores at most n^O(log s)nodes, yielding a quasi-polynomial bound on the size of the search tree in typical instances, where n denotes the decision dimension and s the number of scenarios. A key ingredient of our analysis is an average-case bound on the integrality gap of the natural linear programming (LP) relaxation. We show that this gap shrinks at rate O((logs log^2 n)/n)with high probability. This result is of independent interest, as it implies that the integrality gap grows only logarithmically with the number of scenarios on average.