# Efficient Computation of Expected Hypervolume Improvement Using Box Decomposition Algorithms

## TL;DR

This paper introduces an efficient algorithm for exactly computing the Expected Hypervolume Improvement in multi-objective Bayesian optimization, significantly reducing computational complexity and enabling faster optimization processes.

## Contribution

The paper presents a novel hypervolume decomposition algorithm that improves the computational complexity of EHVI calculation from quadratic and cubic to near-linear, applicable to higher dimensions.

## Key findings

- Algorithm reduces EHVI computation time significantly.
- Complexity improved to Θ(n log n) for two-objective problems.
- Applicable to higher dimensions with hyperbox decomposition.

## Abstract

In the field of multi-objective optimization algorithms, multi-objective Bayesian Global Optimization (MOBGO) is an important branch, in addition to evolutionary multi-objective optimization algorithms (EMOAs). MOBGO utilizes Gaussian Process models learned from previous objective function evaluations to decide the next evaluation site by maximizing or minimizing an infill criterion. A common criterion in MOBGO is the Expected Hypervolume Improvement (EHVI), which shows a good performance on a wide range of problems, with respect to exploration and exploitation. However, so far it has been a challenge to calculate exact EHVI values efficiently. In this paper, an efficient algorithm for the computation of the exact EHVI for a generic case is proposed. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities are improved from previously $O (n^2)$ and $O(n^3)$, for two- and three-objective problems respectively, to $\Theta(n\log n)$, which is asymptotically optimal. This article generalizes the scheme in higher dimensional case by utilizing a new hyperbox decomposition technique, which was proposed by D{\"a}chert et al, EJOR, 2017. It also utilizes a generalization of the multilayered integration scheme that scales linearly in the number of hyperboxes of the decomposition. The speed comparison shows that the proposed algorithm in this paper significantly reduces computation time. Finally, this decomposition technique is applied in the calculation of the Probability of Improvement (PoI).