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. 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 commonly used 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. This paper proposes an efficient algorithm for the exact calculation of the EHVI for in a generic case. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities can be improved from previously O (n^2) and O(n^3), for two- and three-objective problems respectively, to varTheta (nlog n), which is asymptotically optimal. This article generalizes the scheme in higher dimensional cases by utilizing a new hyperbox decomposition technique, which is proposed by Dächert et al. (Eur J Oper Res 260(3):841–855, 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).
Highlights
In multi-objective design optimization, the objective function evaluations are generally computationally costly, mainly due to the long convergence times of simulation models
Compared to evolutionary multi-objective optimization algorithms (EMOAs), Multi-objective Bayesian global optimization (MOBGO) requires only a small budget of function evaluations to achieve a similar result with respect to hypervolume indicator, and it has already been used in real-world applications to solve expensive evaluation problems [40]
The paper is structured as follows: Sect. 2 introduces the nomenclature, Kriging, and the framework of MOBGO; Sect. 3 provides some fundamental definitions used in this paper; Sect. 4 describes how to partition an integration space intoboxes efficiently, and how to calculate Expected Hypervolume Improvement (EHVI) based on this partitioning method; Sect. 5 shows experimental results of speed comparison and MOBGO based algorithms’ performance on 10 well-known scientific benchmarks in 6- and 18-dimensional search spaces; Sect. 6 draws the main conclusions and discusses some potential topics for further research
Summary
In multi-objective design optimization, the objective function evaluations are generally computationally costly, mainly due to the long convergence times of simulation models. A simple and common remedy to this problem is to use a statistical model learned from previous evaluations as the fitness function, instead of the ‘true’ objective function This method is known as Bayesian Global Optimization (BGO) [30]. The first exact EHVI calculation algorithm in the 2-D case was derived in [12], with time complexity of O(n3 log n). Couckuyt et al [5] introduced an exact EHVI calculation algorithm (CDD13) for d > 2 by representing a non-dominated space with three types of boxes, where d represents the number of objective functions. Yang et al proposed an asymptotically optimal algorithm with time complexity Θ(n log n) in the 3-D case [38]. The paper is structured as follows: Sect. 2 introduces the nomenclature, Kriging, and the framework of MOBGO; Sect. 3 provides some fundamental definitions used in this paper; Sect. 4 describes how to partition an integration space into (hyper)boxes efficiently, and how to calculate EHVI based on this partitioning method; Sect. 5 shows experimental results of speed comparison and MOBGO based algorithms’ performance on 10 well-known scientific benchmarks in 6- and 18-dimensional search spaces; Sect. 6 draws the main conclusions and discusses some potential topics for further research
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
