ABSTRACT
We consider a conically constrained optimisation problem where the optimal solution lies at the apex of the cone and study the behaviour of a (1,λ)-ES that handles constraints by resampling infeasible candidate solutions. Expressions that describe the strategy's single-step behaviour are derived. Assuming that the mutation strength is adapted in a scale-invariant manner, a simple zeroth-order model is used to determine the speed of convergence of the strategy. We then derive expressions that approximately characterise the step size and convergence rate attained when using cumulative step size adaptation and compare the values with optimal ones.
- D. V. Arnold. Noisy Optimization with Evolution Strategies. Kluwer Academic Publishers, 2002. Google ScholarDigital Library
- D. V. Arnold. Analysis of a repair mechanism for the (1,ł)-ES applied to a simple constrained problem. In Genetic and Evolutionary Computation Conference -- GECCO 2011, pages 853--860. ACM Press, 2011. Google ScholarDigital Library
- D. V. Arnold. On the behaviour of the (1,ł)-ES for a simple constrained problem. In H.-G. Beyer and W. B. Langdon, editors, Foundations of Genetic Algorithms -- FOGA 2011, pages 15--24. ACM Press, 2011. Google ScholarDigital Library
- D. V. Arnold. On the behaviour of the (1,ł)-σSA-ES for a constrained linear problem. In C. A. Coello Coello et al., editors, Parallel Problem Solving from Nature -- PPSN XII, pages 82--91. Springer Verlag, 2012. Google ScholarDigital Library
- D. V. Arnold. Resampling versus repair in evolution strategies applied to a constrained linear problem. Evolutionary Computation, 2013. to appear.Google Scholar
- D. V. Arnold and H.-G. Beyer. On the behaviour of evolution strategies optimising cigar functions. Evolutionary Computation, 18:661--682, 2010. Google ScholarDigital Library
- D. V. Arnold and D. Brauer. On the behaviour of the $(1Google Scholar
- 1)$-ES for a simple constrained problem. In G. Rudolph et al., editors, Parallel Problem Solving from Nature -- PPSN X, pages 1--10. Springer Verlag, 2008.Google Scholar
- A. Auger and N. Hansen. Reconsidering the progress rate theory for evolution strategies in finite dimensions. In Genetic and Evolutionary Computation Conference -- GECCO 2006, pages 445--452. ACM Press, 2006. Google ScholarDigital Library
- N. Balakrishnan and C. R. Rao. Order statistics: An introduction. In N. Balakrishnan et al., editors, Handbook of Statistics, volume 16, pages 3--24. Elsevier, 1998.Google Scholar
- H.-G. Beyer. Ein Evolutionsverfahren zur mathematischen Modellierung station\"arer Zust\"ande in dynamischen Systemen. PhD thesis, Hochschule für Architektur und Bauwesen, Weimar, 1989.Google Scholar
- H.-G. Beyer and H.-P. Schwefel. Evolution strategies -- A comprehensive introduction. Natural Computing, 1(1):3--52, 2002. Google ScholarDigital Library
- N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 9(2):159--195, 2001. Google ScholarDigital Library
- Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1--32, 1996. Google ScholarDigital Library
- A. Ostermeier, A. Gawelczyk, and N. Hansen. Step-size adaptation based on non-local use of selection information. In Y. Davidor et al., editors, Parallel Problem Solving from Nature -- PPSN III, pages 189--198. Springer Verlag, 1994. Google ScholarDigital Library
- A. I. Oyman, K. Deb, and H.-G. Beyer. An alternative constraint handling method for evolution strategies. In Proceedings of the 1999 IEEE Congress on Evolutionary Computation, pages 612--619. IEEE Press, 1999.Google ScholarCross Ref
- I. Rechenberg. Evolutionsstrategie -- Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. Friedrich Frommann Verlag, 1973.Google Scholar
- H.-P. Schwefel. Numerical Optimization of Computer Models. Wiley, 1981. Google ScholarDigital Library
- H.-P. Schwefel. Evolution and Optimum Seeking. Wiley, 1995.Google ScholarDigital Library
Index Terms
- On the behaviour of the (1, λ)-es for a conically constrained problem
Recommendations
A (1+1)-CMA-ES for constrained optimisation
GECCO '12: Proceedings of the 14th annual conference on Genetic and evolutionary computationThis paper introduces a novel constraint handling approach for covariance matrix adaptation evolution strategies (CMA-ES). The key idea is to approximate the directions of the local normal vectors of the constraint boundaries by accumulating steps that ...
On the behaviour of the (1, λ)-es for conically constrained linear problems
We study the behaviour of a <inline-formula><inline-graphic xlink="EVCO_a_00125inline1.gif" xlink:type="simple"/></inline-formula>-ES that handles constraints by resampling infeasible candidate solutions for linear optimization problems with a conically ...
Reconsidering constraint release for active-set evolution strategies
GECCO '17: Proceedings of the Genetic and Evolutionary Computation ConferenceWe consider the problem of solving constrained numerical optimization problems where the objective function is a black box, but the constraint functions are known explicitly. A recently proposed active-set approach implemented in an evolution strategy ...
Comments