A sampling criterion for constrained Bayesian optimization with uncertainties

TL;DR

This paper introduces a novel Bayesian optimization method for chance constrained problems affected by uncertainties, focusing on an acquisition criterion that balances objective improvement and constraint reliability, with demonstrated efficiency through numerical tests.

Contribution

It proposes a new acquisition criterion for constrained Bayesian optimization under uncertainty, derived from Stepwise Uncertainty Reduction, with analytical expressions for efficient computation.

Findings

The new criterion improves optimization efficiency compared to alternatives.

Analytical expressions enable fast computation of the acquisition function.

Numerical studies validate the effectiveness of the proposed method.

Abstract

We consider the problem of chance constrained optimization where it is sought to optimize a function and satisfy constraints, both of which are affected by uncertainties. The real world declinations of this problem are particularly challenging because of their inherent computational cost. To tackle such problems, we propose a new Bayesian optimization method. It applies to the situation where the uncertainty comes from some of the inputs, so that it becomes possible to define an acquisition criterion in the joint controlled-uncontrolled input space. The main contribution of this work is an acquisition criterion that accounts for both the average improvement in objective function and the constraint reliability. The criterion is derived following the Stepwise Uncertainty Reduction logic and its maximization provides both optimal controlled and uncontrolled parameters. Analytical expressions are given to efficiently calculate the criterion. Numerical studies on test functions are presented. It is found through experimental comparisons with alternative sampling criteria that the adequation between the sampling criterion and the problem contributes to the efficiency of the overall optimization. As a side result, an expression for the variance of the improvement is given.