Expected hypervolume improvement for simultaneous multi-objective and multi-fidelity optimization

Bayesian optimization has proven to be an efficient method to optimize expensive-to-evaluate systems. However, depending on the cost of single observations, multi-dimensional optimizations of one or more objectives may still be prohibitively expensive. Multi-fidelity optimization remedies this issue by including multiple, cheaper information sources such as low-resolution approximations in numerical simulations. Acquisition functions for multi-fidelity optimization are typically based on exploration-heavy algorithms that are difficult to combine with optimization towards multiple objectives. Here we show that the expected hypervolume improvement policy can act in many situations as a suitable substitute. We incorporate the evaluation cost either via a two-step evaluation or within a single acquisition function with an additional fidelity-related objective. This permits simultaneous multi-objective and multi-fidelity optimization, which allows to accurately establish the Pareto set and front at fractional cost. Benchmarks show a cost reduction of an order of magnitude or more. Our method thus allows for Pareto optimization of extremely expansive black-box functions. The presented methods are simple and straightforward to implement in existing, optimized Bayesian optimization frameworks and can immediately be extended to batch optimization. The techniques can also be used to combine different continuous and/or discrete fidelity dimensions, which makes them particularly relevant for simulation problems in plasma physics, fluid dynamics and many other branches of scientific computing.