Faster Computation of Expected Hypervolume Improvement

Hupkens, Iris; Emmerich, Michael; Deutz, André H.

doi:10.48550/arxiv.1408.7114

Cited by 2 publications

(5 citation statements)

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“…The integration of ( 12) over B \ H n , in the expression (11) of the multi-objective EI criterion, is a nontrivial problem. Several authors (Emmerich and Klinkenberg, 2008;Bader and Zitzler, 2011;Hupkens et al, 2014;Couckuyt et al, 2014) have proposed decomposition methods to carry out this computation, where the integration domain B \ H n is partitioned into hyper-rectangles, over which the integral can be computed analytically. The computational complexity of these methods, however, increases exponentially with the number of objectives 3 , which makes the approach impractical in problems with more than a few objective functions.…”

Section: Remarkmentioning

confidence: 99%

“…Since the dimension of Y is likely to be high in practical problems (say, p + q ≥ 5), the integration of y → P n (ξ(x) y) over G n cannot be carried out using decomposition methods (Emmerich and Klinkenberg, 2008;Bader and Zitzler, 2011;Hupkens et al, 2014) because, as mentioned in Section 2.2, the computational complexity of these methods increases exponentially with the dimension of Y.…”

Section: Computation Of the Expected Improvementmentioning

confidence: 99%

“…Dash symbols are used when a value cannot be calculated. criterion since exact and efficient domain-decomposition-based algorithms are available (see Hupkens et al, 2014;Couckuyt et al, 2014, and references therein). An in-depth study of the quality of the approximation provided by our SMC method, and a comparison with exact methods, is therefore needed before more precise recommandations can be made.…”

Section: Multi-objective Optimization Benchmarkmentioning

confidence: 99%

“…This is the most common modeling assumption in the Bayesian optimization literature, when several objective functions, and possibly also several constraint functions, have to be dealt with. See the VIPER algorithm ofWilliams et al (2010) for an example of an algorithm based on correlated Gaussian processes.3 See, e.g.,Beume (2009),Hupkens et al (2014),Couckuyt et al (2014) and references therein for decomposition algorithms and complexity results.…”

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confidence: 99%

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A Bayesian approach to constrained single- and multi-objective optimization

2016

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This article addresses the problem of derivative-free (single-or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria-the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions-as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single-and multi-objective constrained optimization.

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Section: Remarkmentioning

confidence: 99%

Section: Computation Of the Expected Improvementmentioning

confidence: 99%

Section: Multi-objective Optimization Benchmarkmentioning

confidence: 99%

mentioning

confidence: 99%

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A Bayesian approach to constrained single- and multi-objective optimization

2016

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“…This involves a sum across all non-dominated cells, whose number grows exponentially with the number of objectives K. In particular, the total number of cells is (| X | + 1) K . Thus, although some methods have been suggested to speed-up its calculation, e.g., Hupkens et al (2014), EHI is only feasible for a 2 or 3 objectives at most.…”

Section: Related Workmentioning

confidence: 99%

Predictive Entropy Search for Multi-objective Bayesian Optimization

Hernández-Lobato¹,

Hernández-Lobato²,

Shah³

et al. 2015

Preprint

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We present PESMO, a Bayesian method for identifying the Pareto set of multi-objective optimization problems, when the functions are expensive to evaluate. The central idea of PESMO is to choose evaluation points so as to maximally reduce the entropy of the posterior distribution over the Pareto set. Critically, the PESMO multi-objective acquisition function can be decomposed as a sum of objectivespecific acquisition functions, which enables the algorithm to be used in decoupled scenarios in which the objectives can be evaluated separately and perhaps with different costs. This decoupling capability also makes it possible to identify difficult objectives that require more evaluations. PESMO also offers gains in efficiency, as its cost scales linearly with the number of objectives, in comparison to the exponential cost of other methods. We compare PESMO with other related methods for multi-objective Bayesian optimization on synthetic and real-world problems. The results show that PESMO produces better recommendations with a smaller number of evaluations of the objectives, and that a decoupled evaluation can lead to improvements in performance, particularly when the number of objectives is large.

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Faster Computation of Expected Hypervolume Improvement

Cited by 2 publications

References 0 publications

A Bayesian approach to constrained single- and multi-objective optimization

A Bayesian approach to constrained single- and multi-objective optimization

Predictive Entropy Search for Multi-objective Bayesian Optimization

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