A multi-recombinative active matrix adaptation evolution strategy for constrained optimization

This paper presents the multi-recombinative constraint active matrix adaptation evolution strategy (Constraint Active-MA-ES). It extends the MA-ES recently introduced by Beyer and Sendhoff in order to handle constrained black-box optimization problems. The active covariance matrix adaptation approach for constraint handling similar to the method proposed by Arnold and Hansen for the $$(1+1)$$(1+1) covariance matrix adaptation evolution strategy is used. As a first step toward constraint handling, active covariance matrix adaptation is incorporated into the MA-ES and evaluated on unconstrained problems (Active-MA-ES). As the second step, constraint handling based on active covariance matrix adaptation for the MA-ES is proposed (Constraint Active-MA-ES). The algorithm has been tested on different test functions and it has been compared to other methods. The experiments show that for cases where directional sampling of the offspring mutations is beneficial, the Active-MA-ES can reach the target faster than the MA-ES. In particular, the Active-MA-ES reaches the final target precision on average by a factor of 1.4 generations (Ellipsoid), a factor of 1.6 generations (Different Powers), and a factor of 2.0 generations (Tablet) faster than the MA-ES. The experiments for the Constraint Active-MA-ES reveal that it achieves 80% of the considered targets with about $$N \times 10^5$$N×105 function and constraint evaluations. With this result, it is the best method among the compared approaches.

[1]  Katya Scheinberg,et al.  Introduction to derivative-free optimization , 2010, Math. Comput..

[2]  Anne Auger,et al.  Linearly Convergent Evolution Strategies via Augmented Lagrangian Constraint Handling , 2017, FOGA '17.

[3]  Xin Yao,et al.  Stochastic ranking for constrained evolutionary optimization , 2000, IEEE Trans. Evol. Comput..

[4]  Dirk V. Arnold,et al.  An Active-Set Evolution Strategy for Optimization with Known Constraints , 2016, PPSN.

[5]  Margaret H. Wright,et al.  Direct search methods: Once scorned, now respectable , 1996 .

[6]  Hans-Paul Schwefel,et al.  Numerical optimization of computer models , 1981 .

[7]  Dirk V. Arnold,et al.  A (1+1)-CMA-ES for constrained optimisation , 2012, GECCO '12.

[8]  Carlos A. Coello Coello,et al.  A simple multimembered evolution strategy to solve constrained optimization problems , 2005, IEEE Transactions on Evolutionary Computation.

[9]  Chukwudi Anyakoha,et al.  A review of particle swarm optimization. Part I: background and development , 2007, Natural Computing.

[10]  Carlos A. Coello Coello,et al.  Constraint-handling in nature-inspired numerical optimization: Past, present and future , 2011, Swarm Evol. Comput..

[11]  Oliver Kramer,et al.  A Review of Constraint-Handling Techniques for Evolution Strategies , 2010, Appl. Comput. Intell. Soft Comput..

[12]  Dirk V. Arnold,et al.  Reconsidering constraint release for active-set evolution strategies , 2017, GECCO.

[13]  Nikolaus Hansen,et al.  The CMA Evolution Strategy: A Tutorial , 2016, ArXiv.

[14]  Anne Auger,et al.  COCO: Performance Assessment , 2016, ArXiv.

[15]  Dirk V. Arnold,et al.  Improving Evolution Strategies through Active Covariance Matrix Adaptation , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[16]  Michael W. Trosset,et al.  I Know It When I See It: Toward a Definition of Direct Search Methods , 1996 .

[17]  W. Vent,et al.  Rechenberg, Ingo, Evolutionsstrategie — Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. 170 S. mit 36 Abb. Frommann‐Holzboog‐Verlag. Stuttgart 1973. Broschiert , 1975 .

[18]  Hans-Georg Beyer,et al.  On the Design of Constraint Covariance Matrix Self-Adaptation Evolution Strategies Including a Cardinality Constraint , 2012, IEEE Transactions on Evolutionary Computation.

[19]  Ponnuthurai N. Suganthan,et al.  Differential evolution with ensemble of constraint handling techniques for solving CEC 2010 benchmark problems , 2010, IEEE Congress on Evolutionary Computation.

[20]  Fang Ming,et al.  Solving constrained optimization problems by using covariance matrix adaptation evolutionary strategy with constraint handling methods , 2018, ICIAI '18.

[21]  Tamara G. Kolda,et al.  Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods , 2003, SIAM Rev..

[22]  Radka Polakova,et al.  L-SHADE with competing strategies applied to constrained optimization , 2017, 2017 IEEE Congress on Evolutionary Computation (CEC).

[23]  Anne Auger,et al.  Markov Chain Analysis of Cumulative Step-Size Adaptation on a Linear Constrained Problem , 2015, Evolutionary Computation.

[24]  Dirk V. Arnold,et al.  Weighted multirecombination evolution strategies , 2006, Theor. Comput. Sci..

[25]  Dirk V. Arnold,et al.  Towards an Augmented Lagrangian Constraint Handling Approach for the (1+1)-ES , 2015, GECCO.

[26]  K. Deb,et al.  An alternative constraint handling method for evolution strategies , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[27]  Bernhard Sendhoff,et al.  Simplify Your Covariance Matrix Adaptation Evolution Strategy , 2017, IEEE Transactions on Evolutionary Computation.

[28]  Chukwudi Anyakoha,et al.  A review of particle swarm optimization. Part II: hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications , 2008, Natural Computing.

[29]  Sebastien Defoort,et al.  Modified Covariance Matrix Adaptation – Evolution Strategy algorithm for constrained optimization under uncertainty, application to rocket design , 2015 .

[30]  Hans-Georg Beyer,et al.  Large Scale Black-Box Optimization by Limited-Memory Matrix Adaptation , 2019, IEEE Transactions on Evolutionary Computation.

[31]  A. Kai Qin,et al.  Self-adaptive Differential Evolution Algorithm for Constrained Real-Parameter Optimization , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[32]  Ali Osman Kusakci,et al.  An adaptive penalty based covariance matrix adaptation-evolution strategy , 2013, Comput. Oper. Res..

[33]  Anne Auger,et al.  Augmented Lagrangian Constraint Handling for CMA-ES - Case of a Single Linear Constraint , 2016, PPSN.

[34]  Youhei Akimoto,et al.  Modified box constraint handling for the covariance matrix adaptation evolution strategy , 2017, GECCO.