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

2024-08-30

arXiv.org

AI Summary

We address expensive, non-convex, black-box two-stage stochastic optimization problems—common in simulation-driven applications—where conventional stochastic programming methods fail due to their reliance on convexity and low-cost function evaluations. Our approach jointly optimizes both first-stage (“here-and-now”) and second-stage (“wait-and-see”) decision variables. We propose the first knowledge-gradient (KG)-based acquisition function for joint variable optimization, accompanied by asymptotic consistency guarantees. To enable practical implementation, we develop an efficient Monte Carlo approximation algorithm. Empirical evaluation across multiple simulation benchmarks demonstrates that our method significantly outperforms standard baselines—including alternating optimization and naive two-step strategies—and surpasses existing state-of-the-art approaches. Moreover, its performance is comparable to more accurate yet computationally intensive surrogate-based alternatives, establishing a new trade-off frontier between accuracy and efficiency.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Optimizes non-convex two-stage stochastic programs.
• Uses Bayesian optimization for expensive evaluations.
• Develops efficient knowledge-gradient acquisition function.

Innovation

• Bayesian optimization for non-convex problems
• Knowledge-gradient-based acquisition function
• Asymptotic consistency guarantee