Bayesian Optimization for Non-Convex Two-Stage Stochastic Optimization Problems

TL;DR

This paper introduces a Bayesian optimization approach tailored for non-convex, two-stage stochastic programming problems that are expensive to evaluate, providing a novel acquisition function and demonstrating superior empirical performance.

Contribution

It develops a knowledge-gradient-based Bayesian optimization method for non-convex two-stage stochastic problems, with theoretical guarantees and efficient approximation.

Findings

Achieves comparable results to less approximate methods

Outperforms existing state-of-the-art approaches

Demonstrates effectiveness on simulation-based objectives

Abstract

Bayesian optimization is a sample-efficient method for solving expensive, black-box optimization problems. Stochastic programming concerns optimization under uncertainty where, typically, average performance is the quantity of interest. In the first stage of a two-stage problem, here-and-now decisions must be made in the face of uncertainty, while in the second stage, wait-and-see decisions are made after the uncertainty has been resolved. Many methods in stochastic programming assume that the objective is cheap to evaluate and linear or convex. We apply Bayesian optimization to solve non-convex, two-stage stochastic programs which are black-box and expensive to evaluate as, for example, is often the case with simulation objectives. We formulate a knowledge-gradient-based acquisition function to jointly optimize the first- and second-stage variables, establish a guarantee of asymptotic consistency, and provide a computationally efficient approximation. We demonstrate comparable empirical results to an alternative we formulate with fewer approximations, which alternates its focus between the two variable types, and superior empirical results over the state of the art and the standard, na\"ive, two-step benchmark.