A Competitive Divide-and-Conquer Algorithm for Unconstrained Large-Scale Black-Box Optimization

This article proposes a competitive divide-and-conquer algorithm for solving large-scale black-box optimization problems for which there are thousands of decision variables and the algebraic models of the problems are unavailable. We focus on problems that are partially additively separable, since this type of problem can be further decomposed into a number of smaller independent subproblems. The proposed algorithm addresses two important issues in solving large-scale black-box optimization: (1) the identification of the independent subproblems without explicitly knowing the formula of the objective function and (2) the optimization of the identified black-box subproblems. First, a Global Differential Grouping (GDG) method is proposed to identify the independent subproblems. Then, a variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is adopted to solve the subproblems resulting from its rotation invariance property. GDG and CMA-ES work together under the cooperative co-evolution framework. The resultant algorithm, named CC-GDG-CMAES, is then evaluated on the CEC’2010 large-scale global optimization (LSGO) benchmark functions, which have a thousand decision variables and black-box objective functions. The experimental results show that, on most test functions evaluated in this study, GDG manages to obtain an ideal partition of the index set of the decision variables, and CC-GDG-CMAES outperforms the state-of-the-art results. Moreover, the competitive performance of the well-known CMA-ES is extended from low-dimensional to high-dimensional black-box problems.

[1]  Xin Yao,et al.  Large scale evolutionary optimization using cooperative coevolution , 2008, Inf. Sci..

[2]  Francisco Herrera,et al.  MA-SW-Chains: Memetic algorithm based on local search chains for large scale continuous global optimization , 2010, IEEE Congress on Evolutionary Computation.

[3]  J. Hopcroft,et al.  Algorithm 447: efficient algorithms for graph manipulation , 1973, CACM.

[4]  George B. Dantzig,et al.  Linear Programming 1: Introduction , 1997 .

[5]  Maliha S. Nash,et al.  Handbook of Parametric and Nonparametric Statistical Procedures , 2001, Technometrics.

[6]  D. Andrews,et al.  Additive Interactive Regression Models: Circumvention of the Curse of Dimensionality , 1990, Econometric Theory.

[7]  Kazuhiro Seki,et al.  Block coordinate descent algorithms for large-scale sparse multiclass classification , 2013, Machine Learning.

[8]  Xiaodong Li,et al.  Cooperative Co-evolution with delta grouping for large scale non-separable function optimization , 2010, IEEE Congress on Evolutionary Computation.

[9]  Tyson R. Browning,et al.  Applying the design structure matrix to system decomposition and integration problems: a review and new directions , 2001, IEEE Trans. Engineering Management.

[10]  Xiaodong Li,et al.  A Comparative Study of CMA-ES on Large Scale Global Optimisation , 2010, Australasian Conference on Artificial Intelligence.

[11]  Jorge Nocedal,et al.  Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization , 1997, TOMS.

[12]  Xiaodong Li,et al.  Cooperative Co-Evolution With Differential Grouping for Large Scale Optimization , 2014, IEEE Transactions on Evolutionary Computation.

[13]  Paul Tseng,et al.  Fortified-Descent Simplicial Search Method: A General Approach , 1999, SIAM J. Optim..

[14]  G. Harik Learning gene linkage to efficiently solve problems of bounded difficulty using genetic algorithms , 1997 .

[15]  Katya Scheinberg,et al.  Recent progress in unconstrained nonlinear optimization without derivatives , 1997, Math. Program..

[16]  Charles Audet,et al.  Mesh Adaptive Direct Search Algorithms for Constrained Optimization , 2006, SIAM J. Optim..

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

[18]  Pedro Larrañaga,et al.  Estimation of Distribution Algorithms , 2002, Genetic Algorithms and Evolutionary Computation.

[19]  Kenneth A. De Jong,et al.  A Cooperative Coevolutionary Approach to Function Optimization , 1994, PPSN.

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

[21]  G. Gary Wang,et al.  Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions , 2010 .

[22]  Peter Richtárik,et al.  Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function , 2011, Mathematical Programming.

[23]  William W. Hager,et al.  Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent , 2006, TOMS.

[24]  C. T. Kelley,et al.  An Implicit Filtering Algorithm for Optimization of Functions with Many Local Minima , 1995, SIAM J. Optim..

[25]  P. Tseng Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization , 2001 .

[26]  C. J. Stone,et al.  Additive Regression and Other Nonparametric Models , 1985 .

[27]  Andries Petrus Engelbrecht,et al.  A Cooperative approach to particle swarm optimization , 2004, IEEE Transactions on Evolutionary Computation.

[28]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[29]  Raymond Ros,et al.  A Simple Modification in CMA-ES Achieving Linear Time and Space Complexity , 2008, PPSN.

[30]  Hongfei Teng,et al.  Cooperative Co-evolutionary Differential Evolution for Function Optimization , 2005, ICNC.

[31]  Luís N. Vicente,et al.  Using Sampling and Simplex Derivatives in Pattern Search Methods , 2007, SIAM J. Optim..

[32]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[33]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[34]  H. H. Rosenbrock,et al.  An Automatic Method for Finding the Greatest or Least Value of a Function , 1960, Comput. J..

[35]  Karl-Heinz Hoffmann,et al.  Optimizing Simulated Annealing , 1990, PPSN.

[36]  Xiaodong Li,et al.  Cooperative Coevolution With Route Distance Grouping for Large-Scale Capacitated Arc Routing Problems , 2014, IEEE Transactions on Evolutionary Computation.

[37]  O. SIAMJ.,et al.  ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS , 1997 .

[38]  J. Hopcroft,et al.  Efficient algorithms for graph manipulation , 1971 .

[39]  Xiaodong Li,et al.  Cooperative Co-evolution for large scale optimization through more frequent random grouping , 2010, IEEE Congress on Evolutionary Computation.

[40]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[41]  F. Wilcoxon Individual Comparisons by Ranking Methods , 1945 .

[42]  Michèle Sebag,et al.  Adaptive coordinate descent , 2011, GECCO '11.

[43]  H. Rabitz,et al.  High Dimensional Model Representations , 2001 .

[44]  Ponnuthurai Nagaratnam Suganthan,et al.  Benchmark Functions for the CEC'2013 Special Session and Competition on Large-Scale Global Optimization , 2008 .

[45]  Sebastian Fischer,et al.  Cognitive Science An Introduction To The Study Of Mind , 2016 .

[46]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[47]  Jim Smith,et al.  An Adaptive Poly-Parental Recombination Strategy , 1995, Evolutionary Computing, AISB Workshop.

[48]  Karsten Weicker,et al.  On the improvement of coevolutionary optimizers by learning variable interdependencies , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[49]  YaoXin,et al.  A Competitive Divide-and-Conquer Algorithm for Unconstrained Large-Scale Black-Box Optimization , 2016 .

[50]  J. Friedenberg,et al.  Cognitive Science: An Introduction to the Study of Mind , 2005 .

[51]  Zhenyu Yang,et al.  Large-Scale Global Optimization Using Cooperative Coevolution with Variable Interaction Learning , 2010, PPSN.

[52]  Jean François Pique,et al.  A revised simplex method with integer Q-matrices , 2001, TOMS.

[53]  Robert Hooke,et al.  `` Direct Search'' Solution of Numerical and Statistical Problems , 1961, JACM.

[54]  Xiaodong Li,et al.  Effective decomposition of large-scale separable continuous functions for cooperative co-evolutionary algorithms , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[55]  Nikolaus Hansen,et al.  Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

[56]  Xin Yao,et al.  Evolutionary programming made faster , 1999, IEEE Trans. Evol. Comput..

[57]  Arnold Neumaier,et al.  Global Optimization by Multilevel Coordinate Search , 1999, J. Glob. Optim..

[58]  J. Friedman,et al.  FLEXIBLE PARSIMONIOUS SMOOTHING AND ADDITIVE MODELING , 1989 .

[59]  Xiaodong Li,et al.  Smart use of computational resources based on contribution for cooperative co-evolutionary algorithms , 2011, GECCO '11.

[60]  F. Glover,et al.  Fundamentals of Scatter Search and Path Relinking , 2000 .

[61]  Russell C. Eberhart,et al.  A new optimizer using particle swarm theory , 1995, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science.

[62]  X. Yao,et al.  Scaling up fast evolutionary programming with cooperative coevolution , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[63]  Xiaodong Li,et al.  Cooperatively Coevolving Particle Swarms for Large Scale Optimization , 2012, IEEE Transactions on Evolutionary Computation.

[64]  Nikolaus Hansen,et al.  The CMA Evolution Strategy: A Comparing Review , 2006, Towards a New Evolutionary Computation.

[65]  Nikolaos V. Sahinidis,et al.  Derivative-free optimization: a review of algorithms and comparison of software implementations , 2013, J. Glob. Optim..

[66]  R. Howard,et al.  Local convergence analysis of a grouped variable version of coordinate descent , 1987 .

[67]  Wojciech Cellary,et al.  Algorithm 520: An Automatic Revised Simplex Method for Constrained Resource Network Scheduling [H] , 1977, TOMS.

[68]  Donald R. Jones,et al.  Direct Global Optimization Algorithm , 2009, Encyclopedia of Optimization.