Bayesian Optimization for Non-Convex Two-Stage Stochastic Optimization Problems (2408.17387v1)

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 this 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. In this work, we apply Bayesian optimization to solve non-convex, two-stage stochastic programs which are expensive to evaluate. 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 which alternates its focus between the two variable types, and superior empirical results over the standard, naive, two-step benchmark. We show that differences in the dimension and length scales between the variable types can lead to inefficiencies of the two-step algorithm, while the joint and alternating acquisition functions perform well in all problems tested. Experiments are conducted on both synthetic and real-world examples.

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

This paper presents a novel application of Bayesian optimization (BO) to non-convex, two-stage stochastic optimization problems where function evaluations are expensive. Stochastic programming deals with optimization under uncertainty and often assumes linear or convex objectives that are cheap to evaluate. However, this work explores scenarios where traditional assumptions do not hold, requiring an efficient and robust optimization strategy. The authors propose a novel knowledge-gradient-based acquisition function tailored for jointly optimizing first- and second-stage variables in two-stage stochastic problems.

Overview

The paper introduces a computationally efficient adaptation of the knowledge gradient (KG) acquisition function to handle the intricacies of non-convex, two-stage stochastic optimization. This approach jointly optimizes the here-and-now decisions in the first stage and the wait-and-see actions in the second stage. It also ensures asymptotic consistency, meaning the algorithm is guaranteed to converge to the optimal solution as the number of samples increases.

Several benchmarks and comparisons are made:

1. Joint Knowledge Gradient (jKG): This method directly addresses the joint optimization of both sets of variables, leveraging a novel acquisition function.
2. Alternating Knowledge Gradient (aKG): Alternates between optimizing the first-stage and second-stage variables using a two-phase strategy, each adopting a specialized KG formulation.
3. Two-Step Knowledge Gradient (2sKG): Sequentially optimizes each stage independently, aiming to find an optimal first-stage decision before addressing the second-stage.

Key Contributions

1. Joint Optimization of Two-Stage Variables: The authors formulated a novel joint KG acquisition function specifically for two-stage stochastic optimization problems. This function enables simultaneous optimization of first- and second-stage variables, addressing inefficiencies observed in two-step optimization methods.
2. Theoretical Foundation: The paper provides a rigorous proof of asymptotic consistency, demonstrating that the proposed joint KG method will, with enough iterations, converge to the optimal solution.
3. Empirical Evaluation: Through extensive experimentation on synthetic and real-world problems, the authors show that the proposed joint KG method outperforms both na\"ive two-step and alternating optimization methods, especially when the underlying problem dimensions and length scales vary between stages.

Experimental Insights

Synthetic Test Problems

The synthetic experiments were designed to evaluate the performance of Bayesian optimization methods under different scenarios:

• Varying Dimensions: The joint KG and alternating KG methods consistently outperformed the two-step and random sampling benchmarks. Performance was robust across various combinations of fixed design ($d_x$), adjustable ($d_y$), and environmental ($d_u$) dimensions.
• Length Scales: The algorithms were tested on Gaussian processes with different length scales. The joint KG method showed superiority in handling cases where different variables had different length scales, benefiting from the simultaneous optimization of first- and second-stage decisions.
• Observation Noise: The ability to handle noisy observations was crucial. The knowledge gradient approaches, particularly joint KG, maintained robustness in the presence of substantial observation noise, significantly outperforming the other methods.

Real-World Example: Optical Table Design

A practical scenario, the design of an optical table for minimizing vibrations, further validated the joint KG method’s efficiency. The system’s differential equations and the harmonic response provided a realistic application demonstrating that the joint KG method could significantly reduce the steady-state amplitude ratio $B/A$ (vibrations of the table $y$ relative to the floor $y_f$) compared to the non-joint alternatives.

Computational Efficiency

While the joint KG method showed superior performance, it is computationally more demanding due to its simultaneous optimization process. However, the paper demonstrates that the time required for optimizing the acquisition function remains negligible relative to the cost of evaluating the expensive black-box function, making the method practical for problems where evaluation time is substantial.

Conclusion and Future Directions

The research significantly contributes to the field of Bayesian optimization by extending it to complex, two-stage stochastic problems. The joint KG acquisition function introduces a more efficient and theoretically grounded method for tackling these problems. Future research could explore extending these approaches to:

• Higher Dimensions: Incorporating techniques from high-dimensional BO to handle larger problems in fields such as wind farm layout optimization.
• Constraints and Multi-Stage Problems: Addressing black-box constraints and extending the methodology to multi-stage stochastic optimization could provide further advancements.
• Diverse Risk Measures: Incorporating risk measures beyond the expected value, such as value-at-risk or mean-variance trade-offs, would broaden the applicability of the proposed methods.

This work lays the groundwork for more robust and efficient stochastic optimization strategies, promising significant improvements in fields requiring complex, high-stakes decision-making under uncertainty.