ParEGO: an algorithm for multiobjective optimization of expensive functions
Supporting material for the papers:
- Knowles, J. (2006) ParEGO: A hybrid algorithm with on-line landscape approximation
for expensive multiobjective optimization problems. IEEE Transactions on Evolutionary Computation. 10 (1): 50-66. Preprint version: PDF (981 KB) Published version from IEEE Explore. BibTeX
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Knowles, J. (2004) ParEGO: A Hybrid Algorithm with On-line Landscape Approximation for Expensive Multiobjective Optimization Problems. Technical Report TR-COMPSYSBIO-2004-01, University of Manchester, Manchester, UK. September 2004. PDF BibTeX
- Knowles, J and Hughes, E. J. (2005)
Multiobjective optimization on a budget of 250 evaluations. Evolutionary Multi-Criterion Optimization (EMO 2005), LNCS 3410, pp. 176-190, Springer-Verlag. PDF BibTeX
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Abstract of the technical report:-
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A great many optimization scenarios,
particularly those arising in the experimental sciences and engineering,
involve fitness evaluations that are expensive
to perform, either financially or temporally, or both. In such applications,
the computational overhead of the optimization algorithm is effectively irrelevant, whereas
the quality of solutions obtained after realistically small numbers
of function evaluations assumes
primary significance, with worst-case
performance, as well as average-case, being especially pertinent. These simple principles apply
equally to multiobjective optimization as to optimization of a single objective function, yet
nearly all multiobjective evolutionary algorithm (MOEA) studies (even some of those relating
to expensive fitness functions) report performance after tens of thousands of evaluations, leaving
open the question of performance on much shorter runs.
In this paper, we concentrate on relatively low-dimensional, non-pathological,
real-valued functions, such as arise in many applications, and assess the performance of two algorithms
at just 100 and 250 function evaluations. The results show that NSGA-II, a popular MOEA,
performs well compared to random search, even under the restricted number of evaluations used.
A better performance (particularly in the worst case) is, however, achieved on our test set
by an algorithm proposed herein - ParEGO - which is a simple extension
of the single-objective efficient global optimization (EGO)
algorithm of Jones et al. ParEGO uses a design-of-experiments
inspired initialization procedure
and learns a Gaussian process model of the search landscape,
which is updated after every function evaluation. Overall, ParEGO exhibits
a promising performance for multiobjective optimization problems where
evaluations are expensive or otherwise restricted in number.
Keywords:- multiobjective optimization, expensive black-box functions, EGO, DACE, NSGA-II, landscape approximation, response surfaces, Pareto optima, test suites, performance assessment
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