A Multicriteria Generalization of Bayesian Global Optimization

Emmerich, Michael; Yang, Kaifeng; Deutz, André H.; Wang, Hao; Fonseca, Carlos M.

doi:10.1007/978-3-319-29975-4_12

Cited by 47 publications

(46 citation statements)

References 17 publications

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“…Hupkens et al [14] found algorithms for computing EHVI with the then lowest worst-case time complexity of O(n 2 ) and O(n 3 ), for two and three objectives, respectively. Recently, Emmerich et al [13] proposed an asymptotically optimal algorithm for the bi-objective case, with a computational time complexity of Θ(n log n). In 3-D, the time complexity of this problem and those of the related problems of computing the Probability of Improvement [8] and Truncated Expected Hypervolume Improvement [2] have so far remained cubic.…”

Section: Relevance and Related Workmentioning

confidence: 99%

“…Then the EHVI for these integration slices will be calculated by the calculation 3d function in Algorithm 1 (line 13 and 23), which is the summation of Eqs. (11), (12), (13) and (14) with the parameters of μ, σ and B 2n+1 . The EHVI computational complexity for each slice is O (1).…”

Section: Computing Slice-based Parts Of the Integralmentioning

confidence: 99%

“…This is a problem, as many SAMCO algorithms require a large number of EHVI computations to be performed in every iteration. Since the announcement of exact computation schemes [12], significant progress has been made, resulting in faster computation schemes [8,13,14]. Notably, in 2-D, the computation time could be improved from O(n 3 log n)-using a straightforward integration scheme, with a grid partitioning and repeated hypervolume computations for each grid cell -to asymptotically optimal Θ(n log n) in [13] using a stripe partitioning and a multi-layered integration scheme.…”

Section: Introductionmentioning

confidence: 99%

“…Since the announcement of exact computation schemes [12], significant progress has been made, resulting in faster computation schemes [8,13,14]. Notably, in 2-D, the computation time could be improved from O(n 3 log n)-using a straightforward integration scheme, with a grid partitioning and repeated hypervolume computations for each grid cell -to asymptotically optimal Θ(n log n) in [13] using a stripe partitioning and a multi-layered integration scheme. However, despite significant efficiency improvements [8,14], the running time of the best-known algorithms for d ≥ 3 has remained in O(n d ).…”

Section: Introductionmentioning

confidence: 99%

“…However, despite significant efficiency improvements [8,14], the running time of the best-known algorithms for d ≥ 3 has remained in O(n d ). This paper shows that the new integration technique proposed in [13] can be generalized to d ≥ 3, yielding a new algorithm with asymptotically optimal time complexity in Θ(n log n). This improves existing upper bounds by a factor of O(n 2 / log n).…”

Section: Introductionmentioning

confidence: 99%

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Computing 3-D Expected Hypervolume Improvement and Related Integrals in Asymptotically Optimal Time

Yang

Emmerich

Deutz

et al. 2017

Lecture Notes in Computer Science

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Abstract. The Expected Hypervolume Improvement (EHVI) is a frequently used infill criterion in surrogate-assisted multi-criterion optimization. It needs to be frequently called during the execution of such algorithms. Despite recent advances in improving computational efficiency, its running time for three or more objectives has remained inwhere d is the number of objective functions and n is the size of the incumbent Pareto-front approximation. This paper proposes a new integration scheme, which makes it possible to compute the EHVI in Θ(n log n) optimal time for the important three-objective case (d = 3). The new scheme allows for a generalization to higher dimensions and for computing the Probability of Improvement (PoI) integral efficiently. It is shown, both theoretically and empirically, that the hidden constant in the asymptotic notation is small. Empirical speed comparisons were designed between the C++ implementations of the new algorithm (which will be in the public domain) and those recently published by competitors, on randomly-generated non-dominated fronts of size 10, 100, and 1000. The experiments include the analysis of batch computations, in which only the parameters of the probability distribution change but the incumbent Pareto-front approximation stays the same. Experimental results show that the new algorithm is always faster than the other algorithms, sometimes over 10 4 times faster.

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Section: Relevance and Related Workmentioning

confidence: 99%

Section: Computing Slice-based Parts Of the Integralmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing 3-D Expected Hypervolume Improvement and Related Integrals in Asymptotically Optimal Time

Yang

Emmerich

Deutz

et al. 2017

Lecture Notes in Computer Science

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Surrogate-assisted multicriteria optimization: Complexities, prospective solutions, and business case

Allmendinger

Emmerich

Hakanen

et al. 2017

J. Multi-Crit. Decis. Anal.

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Complexity in solving real-world multicriteria optimization problems often stems from the fact that complex, expensive, and/or time-consuming simulation tools or physical experiments are used to evaluate solutions to a problem. In such settings, it is common to use efficient computational models, often known as surrogates or metamodels, to approximate the outcome (objective or constraint function value) of a simulation or physical experiment. The presence of multiple objective functions poses an additional layer of complexity for surrogate-assisted optimization.For example, complexities may relate to the appropriate selection of metamodels for the individual objective functions, extensive training time of surrogate models, or the optimal use of many-core computers to approximate efficiently multiple objectives simultaneously. Thinking out of the box, complexity can also be shifted from approximating the individual objective functions to approximating the entire Pareto front. This leads to further complexities, namely, how to validate statistically and apply the techniques developed to real-world problems. In this paper, we discuss emerging complexity-related topics in surrogate-assisted multicriteria optimization that may not be prevalent in nonsurrogate-assisted single-objective optimization. These complexities are motivated using several real-world problems in which the authors were involved. We then discuss several promising future research directions and prospective solutions to tackle emerging complexities in surrogate-assisted multicriteria optimization. Finally, we provide insights from an industrial point of view into how surrogate-assisted multicriteria optimization techniques can be developed and applied within a collaborative business environment to tackle real-world problems.

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