Stochastic Cutting Planes for Data-Driven Optimization

TL;DR

This paper proposes a stochastic cutting-plane method for data-driven MINLO problems, demonstrating high-probability convergence and significant speedups over traditional methods through numerical experiments.

Contribution

It introduces a stochastic variant of the cutting-plane algorithm with theoretical convergence guarantees for data-driven MINLO problems.

Findings

Achieves high-probability convergence to $\epsilon$-optimal solutions.

Delivers multiple order-of-magnitude speedup over standard methods.

Shows sampling size of $O( oot3 ext{n})$ suffices for high-quality solutions.

Abstract

We introduce a stochastic version of the cutting-plane method for a large class of data-driven Mixed-Integer Nonlinear Optimization (MINLO) problems. We show that under very weak assumptions the stochastic algorithm is able to converge to an $\epsilon$-optimal solution with high probability. Numerical experiments on several problems show that stochastic cutting planes is able to deliver a multiple order-of-magnitude speedup compared to the standard cutting-plane method. We further experimentally explore the lower limits of sampling for stochastic cutting planes and show that for many problems, a sampling size of $O(\sqrt[3]{n})$ appears to be sufficient for high quality solutions.