Adaptive Ranking-Based Constraint Handling for Explicitly Constrained Black-Box Optimization

A novel explicit constraint handling technique for the covariance matrix adaptation evolution strategy (CMA-ES) is proposed. The proposed constraint handling exhibits two invariance properties. One is the invariance to arbitrary element-wise increasing transformation of the objective and constraint functions. The other is the invariance to arbitrary affine transformation of the search space. The proposed technique virtually transforms a constrained optimization problem into an unconstrained optimization problem by considering an adaptive weighted sum of the ranking of the objective function values and the ranking of the constraint violations that are measured by the Mahalanobis distance between each candidate solution to its projection onto the boundary of the constraints. Simulation results are presented and show that the CMA-ES with the proposed constraint handling exhibits the affine invariance and performs similarly to the CMA-ES on unconstrained counterparts.

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

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

[3]  Youhei Akimoto,et al.  Natural Gradient Approach for Linearly Constrained Continuous Optimization , 2014, PPSN.

[4]  Sébastien Le Digabel,et al.  A Taxonomy of Constraints in Simulation-Based Optimization , 2015, 1505.07881.

[5]  Oswin Krause,et al.  A CMA-ES with Multiplicative Covariance Matrix Updates , 2015, GECCO.

[6]  Youhei Akimoto,et al.  PSA-CMA-ES: CMA-ES with population size adaptation , 2018, GECCO.

[7]  Petros Koumoutsakos,et al.  A Method for Handling Uncertainty in Evolutionary Optimization With an Application to Feedback Control of Combustion , 2009, IEEE Transactions on Evolutionary Computation.

[8]  Bernhard Sendhoff,et al.  Covariance Matrix Adaptation Revisited - The CMSA Evolution Strategy - , 2008, PPSN.

[9]  Hans-Georg Beyer,et al.  A Covariance Matrix Self-Adaptation Evolution Strategy for Optimization Under Linear Constraints , 2018, IEEE Transactions on Evolutionary Computation.

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

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

[12]  Anne Auger,et al.  Quality Gain Analysis of the Weighted Recombination Evolution Strategy on General Convex Quadratic Functions , 2016, FOGA '17.

[13]  Jing J. Liang,et al.  Problem Deflnitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization , 2006 .

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

[15]  Anne Auger,et al.  Impacts of invariance in search: When CMA-ES and PSO face ill-conditioned and non-separable problems , 2011, Appl. Soft Comput..

[16]  Afonso C. C. Lemonge,et al.  A rank-based constraint handling technique for engineering design optimization problems solved by genetic algorithms , 2017 .

[17]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

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

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

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

[21]  Hans-Georg Beyer,et al.  A multi-recombinative active matrix adaptation evolution strategy for constrained optimization , 2019, Soft Comput..

[22]  Petros Koumoutsakos,et al.  Reducing the Time Complexity of the Derandomized Evolution Strategy with Covariance Matrix Adaptation (CMA-ES) , 2003, Evolutionary Computation.

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

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

[25]  P. Suganthan,et al.  Problem Definitions and Evaluation Criteria for the CEC 2010 Competition on Constrained Real- Parameter Optimization , 2010 .

[26]  John L. Nazareth,et al.  Introduction to derivative-free optimization , 2010, Math. Comput..

[27]  Anne Auger,et al.  Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB-2009 , 2010, GECCO '10.

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

[29]  Y. Fung,et al.  A Theory of Elasticity of the Lung , 1974 .

[30]  Carlos Artemio Coello-Coello,et al.  Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art , 2002 .

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