Package: pypop7
PyPop7 (a Pure-PYthon library of POPulation-based black-box OPtimization)PyPop7is aPure-PYthonlibrary ofPOPulation-based OPtimizationfor single-objective, real-parameter, black-box problems (currently actively maintained). Its goal is to provide aunifiedinterface andelegantimplementations forBlack-Box Optimization (BBO),particularly population-based optimizers, in order to facilitate research repeatability, benchmarking of BBO, and real-world applications.More specifically, for alleviating theircurse of dimensionality, the primary focus ofPyPop7is to cover theirState Of The Art for Large-Scale Optimization (LSO), though many of their small/medium-scaled versions and variants are also included here (mainly for theoretical or benchmarking purposes).How to Quickly Use PyPop7The following three steps are enough to utilize the optimization power of this libraryPyPop7:Usepipto installpypop7on the Python3-based virtual environment viavenvorconda(a strong suggestion):$pipinstallpypop7For simplicity, all required library dependencies (except special cases) areautomaticallyinstalled according tosetup.cfg.Define the objective/cost function (calledfitness functionin this library) for the optimization problem at hand,Run one or more black-box optimizers on this optimization problem:importnumpyasnp# for numerical computation, which is also the computing engine of pypop7# 2. Define your own objective/cost function for the optimization problem at hand:#   the below example is Rosenbrock, the notorious test function from the optimization communitydefrosenbrock(x):return100.0*np.sum(np.power(x[1:]-np.power(x[:-1],2),2))+np.sum(np.power(x[:-1]-1,2))# define the fitness (cost) function and also its settingsndim_problem=1000problem={'fitness_function':rosenbrock,# cost function'ndim_problem':ndim_problem,# dimension'lower_boundary':-5.0*np.ones((ndim_problem,)),# search boundary'upper_boundary':5.0*np.ones((ndim_problem,))}# 3. Run one or more black-box optimizers on the given optimization problem:#   here we choose LM-MA-ES owing to its low complexity and metric-learning ability for LSO#   https://pypop.readthedocs.io/en/latest/es/lmmaes.htmlfrompypop7.optimizers.es.lmmaesimportLMMAES# define all the necessary algorithm options (which differ among different optimizers)options={'fitness_threshold':1e-10,# terminate when the best-so-far fitness is lower than this threshold'max_runtime':3600,# 1 hours (terminate when the actual runtime exceeds it)'seed_rng':0,# seed of random number generation (which must be explicitly set for repeatability)'x':4.0*np.ones((ndim_problem,)),# initial mean of search (mutation/sampling) distribution'sigma':3.0,# initial global step-size of search distribution (not necessarily optimal)'verbose':500}lmmaes=LMMAES(problem,options)# initialize the optimizerresults=lmmaes.optimize()# run its (time-consuming) search processprint(results)Note that forPyPop7, the number7is added just becausepypophas been registered byotherinPyPI. The iconbutterflyforPyPop7is used to respect to the book (a complete variorum edition) ofFisher,"the greatest of Darwin's successors":The Genetical Theory of Natural Selection(where four butterflies were drawn in its cover), which inspired the proposal ofGenetic Algorithms (GA).For a list ofpublic use casesofPyPop7, seethis online documentfor more details. For new/missed black-box optimizers, we provide aunifiedAPI interface to freely add them if they satisfy the following design philosophy (seedevelopment-guidefor details).A Large Number of Black-Box Optimizers (BBO)Note that Ant Colony Optimization (ACO) and Tabu Search (TS) are not covered in this open-source library, since they work mainly indiscrete/combinatorialsearch spaces. Furthermore, brute-force search (exhaustive/grid search) is also excluded here, since it works only forvery low(typically < 10) dimensions. In the near future version, we will consider addingSimultaneous Perturbation Stochastic Approximation (SPSA)into this open-source library.: indicates the specific BBO version for LSO (dimension >= 1000).: indicates the competitive (orde facto) BBO version forsmall/medium-dimensionalproblems (though it may work well under certain LSO circumstances).: indicates the baseline BBO version mainly fortheoreticalinterest, owing to its simplicity (relatively ease to mathematical analysis).Note that this classification based on only the dimension of objective function is just a rough estimation for algorithm selection.In practice, perhaps the simplest way to algorithm selection
is trial-and-error or to try more advancedAutomated Algorithm Selectiontechniques.Evolution Strategies (ES)[e.g.,Ollivier et al., 2017, JMLR;Hansen et al., 2015;Bäck et al., 2013;Rudolph, 2012;Beyer&Schwefel, 2002;Rechenberg, 1989;Schwefel, 1984]Mixture Model-based Evolution Strategy (MMES) [He et al., 2021, TEVC]Fast Covariance Matrix Adaptation Evolution Strategy (FCMAES) [Li et al., 2020, TCYB;Li&Zhang, 2016, PPSN]Limited-Memory Matrix Adaptation Evolution Strategy (LMMAES) [Loshchilov et al., 2019, TEVC]Limited Memory Covariance Matrix Adaptation (LMCMA) [Loshchilov, 2017, ECJ]Limited Memory Covariance Matrix Adaptation Evolution Strategy (LMCMAES) [Loshchilov, 2014, GECCO]Rank-M Evolution Strategy (RMES) [Li&Zhang, 2018, TEVC]Rank-One Evolution Strategy (R1ES) [Li&Zhang, 2018, TEVC]Projection-based Covariance Matrix Adaptation (VKDCMA) [Akimoto&Hansen, 2016, PPSN;Akimoto&Hansen, 2016, GECCO]Linear Covariance Matrix Adaptation (VDCMA) [Akimoto et al., 2014, GECCO]Cholesky-CMA-ES-2016 (CCMAES2016) [Krause et al., 2016, NeurIPS](1+1)-Active-Cholesky-CMA-ES-2015 (OPOA2015) [Krause&Igel, 2015, FOGA](1+1)-Active-Cholesky-CMA-ES (OPOA2010) [Arnold&Hansen, 2010, GECCO]Cholesky-CMA-ES (CCMAES2009) [Suttorp et al., 2009, MLJ](1+1)-Cholesky-CMA-ES-2009 (OPOC2009) [Suttorp et al., 2009, MLJ](1+1)-Cholesky-CMA-ES (OPOC2006) [Igel et al., 2006, GECCO]Separable Covariance Matrix Adaptation Evolution Strategy (SEPCMAES) [Bäck et al., 2013;Ros&Hansen, 2008, PPSN]Diagonal Decoding Covariance Matrix Adaptation (DDCMA) [Akimoto&Hansen, 2020, ECJ]Matrix Adaptation Evolution Strategy (MAES) [Beyer&Sendhoff, 2017, TEVC]Fast Matrix Adaptation Evolution Strategy (FMAES) [Beyer, 2020, GECCO;Loshchilov et al., 2019, TEVC]Covariance Matrix Adaptation Evolution Strategy (CMAES) [e.g.Hansen, 2016;Hansen et al., 2003, ECJ;Hansen et al., 2001, ECJ;Hansen&Ostermeier, 1996, CEC]Self-Adaptation Matrix Adaptation Evolution Strategy (SAMAES) [Beyer, 2020, GECCO]Self-Adaptation Evolution Strategy (SAES) [e.g.Beyer, 2020, GECCO;Beyer, 2007, Scholarpedia]Cumulative Step-size Adaptation Evolution Strategy (CSAES)  [e.g.Hansen et al., 2015;Ostermeier et al., 1994, PPSN]Derandomized Self-Adaptation Evolution Strategy (DSAES) [e.g.Hansen et al., 2015;Ostermeier et al., 1994, ECJ]Schwefel's Self-Adaptation Evolution Strategy (SSAES) [e.g.Hansen et al., 2015;Beyer&Schwefel, 2002;Schwefel, 1988;Schwefel, 1984, AOR]Rechenberg's (1+1)-Evolution Strategy with 1/5th success rule (RES) [e.g.Hansen et al., 2015;Kern et al., 2004;Rechenberg, 1989;Rechenberg, 1984;Schumer&Steiglitz, 1968, TAC]Natural Evolution Strategies (NES)[e.g.Wierstra et al., 2014, JMLR;Yi et al., 2009, ICML;Wierstra et al., 2008, CEC]Rank-One Natural Evolution Strategy (R1NES) [Sun et al., 2013, GECCO]Separable Natural Evolution Strategy (SNES) [Schaul et al., 2011, GECCO]Exponential Natural Evolution Strategies (XNES) [e.g.Glasmachers et al., 2010, GECCO]Exact Natural Evolution Strategy (ENES) [e.g.Sun et al., 2009, ICML]Original Natural Evolution Strategy (ONES) [e.g.Wierstra et al., 2008, CEC]Search Gradient-based Evolution Strategy (SGES) [e.g.Wierstra et al., 2008, CEC]Estimation of Distribution Algorithms (EDA)[e.g.Brookes et al., 2020, GECCO;Larrañaga&Lozano, 2002;Pelikan et al., 2002;Mühlenbein&Paaß, 1996, PPSN;Baluja&Caruana, 1995, ICML]Random-Projection Estimation of Distribution Algorithm (RPEDA) [Kabán et al., 2016, ECJ]Univariate Marginal Distribution Algorithm (UMDA) [e.g.Larrañaga&Lozano, 2002;Mühlenbein, 1997, ECJ]Adaptive Estimation of Multivariate Normal Algorithm (AEMNA)[Larrañaga&Lozano, 2002]Estimation of Multivariate Normal Algorithm (EMNA) [e.g.Larrañaga&Lozano, 2002]Cross-Entropy Method (CEM)[e.g.Rubinstein&Kroese, 2016;Hu et al., 2007, OR;Kroese et al., 2006, MCAP;De Boer et al., 2005, AOR;Rubinstein&Kroese, 2004]Differentiable Cross-Entropy Method (DCEM) [Amos&Yarats, 2020, ICML]Dynamic-Smoothing Cross-Entropy Method (DSCEM) [e.g.Kroese et al., 2006, MCAP]Model Reference Adaptive Search (MRAS) [e.g.Hu et al., 2007, OR]Standard Cross-Entropy Method (SCEM) [e.g.Kroese et al., 2006, MCAP]Differential Evolution (DE)[e.g.Price, 2013;Price et al., 2005;Storn&Price, 1997, JGO]Success-History based Adaptive Differential Evolution (SHADE) [Tanabe&Fukunaga, 2013, CEC]Adaptive Differential Evolution (JADE) [Zhang&Sanderson, 2009, TEVC]Composite Differential Evolution (CODE) [Wang et al., 2011, TEVC]Trigonometric-mutation Differential Evolution (TDE) [Fan&Lampinen, 2003, JGO]Classic Differential Evolution (CDE) [e.g.Storn&Price, 1997, JGO]Particle Swarm Optimizer (PSO)[e.g.Fornasier et al., 2021, JMLR;Bonyadi&Michalewicz, 2017, ECJ;Rahmat-Samii et al., 2012, PIEEE;Escalante et al., 2009, JMLR;Dorigo et al., 2008;Poli et al., 2007, SI;Shi&Eberhart, 1998, CEC;Kennedy&Eberhart, 1995, ICNN]Cooperative Coevolving Particle Swarm Optimizer (CCPSO2) [Li&Yao, 2012, TEVC]Incremental Particle Swarm Optimizer (IPSO) [de Oca et al., 2011, TSMCB]Comprehensive Learning Particle Swarm Optimizer (CLPSO) [Liang et al., 2006, TEVC]Cooperative Particle Swarm Optimizer (CPSO) [Van den Bergh&Engelbrecht, 2004, TEVC]Standard Particle Swarm Optimizer with a Local (ring) topology (SPSOL) [e.g.Shi&Eberhart, 1998, CEC;Kennedy&Eberhart, 1995, ICNN]Standard Particle Swarm Optimizer with a global topology (SPSO) [e.g.Shi&Eberhart, 1998, CEC;Kennedy&Eberhart, 1995, ICNN]Cooperative Co-evolution (CC)[e.g.Gomez et al., 2008, JMLR;Panait et al., 2008, JMLR;Moriarty&Miikkulainen, 1995, ICML;Potter&De Jong, 1994, PPSN]Hierarchical Cooperative Co-evolution (HCC) [Mei et al., 2016, ACM-TOMS;Gomez&Schmidhuber, 2005, ACM-GECCO]CoOperative CO-evolutionary Covariance Matrix Adaptation (COCMA) [Mei et al., 2016, ACM-TOMS;Potter&De Jong, 1994, PPSN]CoOperative co-Evolutionary Algorithm (COEA) [e.g.Panait et al., 2008, JMLR;Potter&De Jong, 1994, PPSN]CoOperative SYnapse NeuroEvolution (COSYNE) [Gomez et al., 2008, JMLR;Moriarty&Miikkulainen, 1995, ICML]Simulated Annealing (SA)[see e.g.Bertsimas&Tsitsiklis, 1993, Statistical Science;Kirkpatrick et al., 1983, Science;Hastings, 1970, Biometrika;Metropolis et al., 1953, JCP]Enhanced Simulated Annealing (ESA) [Siarry et al., 1997, ACM-TOMS]Corana et al.' Simulated Annealing (CSA) [Corana et al., 1987, ACM-TOMS]Noisy Simulated Annealing (NSA) [Bouttier&Gavra, 2019, JMLR]Genetic Algorithms (GA)[e.g.Forrest, 1993, Science;Holland, 1973, SICOMP;Holland, 1962, JACM]Active Subspace Genetic Algorithm (ASGA) [Demo et al., 2021, SISC]Global and Local genetic algorithm (GL25) [García-Martínez et al., 2008, EJOR]Generalized Generation Gap with Parent-Centric Recombination (G3PCX) [Deb et al., 2002, ECJ]GENetic ImplemenTOR (GENITOR) [e.g.Whitley et al., 1993, MLJ]Evolutionary Programming (EP)[e.g.Yao et al., 1999, TEVC;Fogel, 1994, Statistics and Computing]Lévy distribution based Evolutionary Programming (LEP) [Lee&Yao, 2004, TEVC]Fast Evolutionary Programming (FEP) [Yao et al., 1999, TEVC]Classical Evolutionary Programming (CEP) [e.g.Yao et al., 1999, TEVC;Bäck&Schwefel, 1993, ECJ]Direct Search (DS)[e.g.Powell, 1998, Acta-Numerica;Wright, 1996;Hooke&Jeeves, 1961, JACM]Powell's search method (POWELL) [SciPy;Powell, 1964, Computer]Generalized Pattern Search (GPS) [Kochenderfer&Wheeler, 2019;Torczon, 1997, SIAM-JO]Nelder-Mead simplex method (NM) [Dean et al., 1975, Science;Nelder&Mead, 1965, Computer]Hooke-Jeeves direct search method (HJ) [Kochenderfer&Wheeler, 2019;Kaupe, 1963, CACM;Hooke&Jeeves, 1961, JACM]Coordinate Search (CS) [Torczon, 1997, SIAM-JO;Fermi&Metropolis, 1952]Random (stochastic) Search (RS)[e.g.Murphy, 2023;Gao&Sener, 2022, ICML;Russell&Norvig, 2021;Nesterov&Spokoiny, 2017, FoCM;Bergstra&Bengio, 2012, JMLR;Schmidhuber et al., 2001;Cvijović&Klinowski, 1995, Science;Rastrigin, 1986;Solis&Wets, 1981, MOOR;Brooks, 1958, OR]BErnoulli Smoothing (BES) [Gao&Sener, 2022, ICML]Gaussian Smoothing (GS) [Nesterov&Spokoiny, 2017, FoCM]Simple Random Search (SRS) [Rosenstein&Barto, 2001, IJCAI]Annealed Random Hill Climber (ARHC) [e.g.Russell&Norvig, 2021;Schaul et al., 2010, JMLR]Random Hill Climber (RHC) [e.g.Russell&Norvig, 2021;Schaul et al., 2010, JMLR]Pure Random Search (PRS) [e.g.Bergstra&Bengio, 2012, JMLR;Schmidhuber et al., 2001;Brooks, 1958, OR]Design PhilosophyGiven a large number of (black-box) optimizers which keep increasing almost every day, we need some (possibly) widely acceptable criteria to select from them, as presented below in details:Respect forBeauty (Elegance)From theproblem-solvingperspective, we empirically prefer to choose the best optimizer for the black-box optimization problem at hand. For the new problem, however, the best optimizer is oftenunknownin advance (when withouta priorknowledge). As a rule of thumb, we need to compare a (often small) set of available/well-known optimizers and finally choose the best one according to some predefined performance criteria. From theacademic researchperspective, however, we prefer so-calledbeautifuloptimizers, though always keeping theNo Free Lunch Theoremsin mind. Typically, the beauty of one optimizer comes from the following attractive features:model novelty,competitive performance on at least one class of real-world problems,theoretical insights (e.g., convergence),clarity/simplicity for understanding and implementation, andrepeatability.If you find any to meet the above standard, welcome to launchissuesorpullsordiscussions. We will consider it to be included in thepypop7library as soon as possible, if possible. Note that anysuperficialimitationto well-established optimizers (i.e.Old Wine in a New Bottle) will beNOTconsidered here. Sometimes, severalvery complexoptimizers could obtain the top rank on some competitions consisting of onlyartificially-constructedbenchmark functions. However, these optimizers may becomeover-skilledon these artifacts. In our opinions, a good optimizer should be elegant andgeneralizable. If there is no persuasive real-world application reported for it, we will not consider anyvery complexoptimizer in this library, in order to avoid the possiblerepeatabilityandoverfittingissues.Respect forDiversityGiven the universality ofblack-box optimization (BBO)in science and engineering, different research communities have designed different optimizers/methods. The type and number of optimizers are continuing to increase as the more powerful optimizers are always desirable for new and more challenging applications. On the one hand, some of these methods may sharemore or lesssimilarities. On the other hand, they may also showsignificantdifferences (w.r.t. motivations / objectives / implementations / communities / practitioners). Therefore, we hope to cover such a diversity from different research communities such as artificial intelligence (particularly machine learningevolutionary computationand
zeroth-order optimization), mathematical optimization/programming (particularly global optimization), operations research / management science, automatic control, electronic engineering, open-source software, physics, chemistry, and others.Respect forOriginalityFor each optimizer included inPyPop7, we expect to give its original/representative reference (sometimes also including its good implementations/improvements). If you find some important references missed, please do NOT hesitate to contact us (and we will be happy to add it if necessary).Respect forRepeatabilityFor randomized search, properly controlling randomness is very crucial to repeat numerical experiments. Here we follow theRandom Samplingsuggestions fromNumPy. In other worlds, you mustexplicitlyset the random seed for each optimizer. For more discussions aboutrepeatabilityfrom machine learning, evolutionary computation, and metaheuristics communities, refer to the following papers, to name a few:Swan, J., Adriaensen, S., Brownlee, A.E., Hammond, K., Johnson, C.G., Kheiri, A., Krawiec, F., Merelo, J.J.,
Minku, L.L., Özcan, E., Pappa, G.L., et al., 2022.Metaheuristics “in the large”. European Journal of Operational Research, 297(2), pp.393-406.López-Ibáñez, M., Branke, J. and Paquete, L., 2021.Reproducibility in evolutionary computation. ACM Transactions on Evolutionary Learning and Optimization, 1(4), pp.1-21.Sonnenburg, S., Braun, M.L., Ong, C.S., Bengio, S., Bottou, L., Holmes, G., LeCunn, Y., Muller, K.R., Pereira, F., Rasmussen, C.E., Ratsch, G., et al., 2007.The need for open source software in machine learning. Journal of Machine Learning Research, 8, pp.2443-2466.Computational EfficiencyFor LSO, computational efficiency is an indispensable performance criterion of DFOin the post-Moore era. To obtain high-performance computation as much as possible,NumPyis heavily used in this library as the base of numerical computation along withSciPy. Sometimes,Numbais also utilized, in order to further accelerate the wall-clock time.ReferencesFor each algorithm family, we provide severalrepresentativeapplications published on sometop-tierjournals and conferences (such as, Nature, Science, PNAS, PRL, JACS, PIEEE, etc.).Derivative-Free Optimization (DFO)Berahas, A.S., Cao, L., Choromanski, K. and Scheinberg, K., 2022.A theoretical and empirical comparison of gradient approximations in derivative-free optimization. Foundations of Computational Mathematics, 22(2), pp.507-560.Porcelli, M. and Toint, P.L., 2022.Exploiting problem structure in derivative free optimization. ACM Transactions on Mathematical Software, 48(1), pp.1-25.Gao, K. and Sener, O., 2022, June.Generalizing Gaussian smoothing for random search. In International Conference on Machine Learning (pp. 7077-7101). PMLR.Kochenderfer, M.J. and Wheeler, T.A., 2019.Algorithms for optimization. MIT Press.Larson, J., Menickelly, M. and Wild, S.M., 2019.Derivative-free optimization methods. Acta Numerica, 28, pp.287-404.Nesterov, Y., 2018.Lectures on convex optimization. Berlin: Springer International Publishing.Nesterov, Y. and Spokoiny, V., 2017.Random gradient-free minimization of convex functions. Foundations of Computational Mathematics, 17(2), pp.527-566.Conn, A.R., Scheinberg, K. and Vicente, L.N., 2009.Introduction to derivative-free optimization. Society for Industrial and Applied Mathematics.Evolutionary Computation (EC) [[Box, 1957]]Eiben, A.E. and Smith, J., 2015.From evolutionary computation to the evolution of things. Nature, 521(7553), pp.476-482. [http://www.evolutionarycomputation.org/]Miikkulainen, R. and Forrest, S., 2021.A biological perspective on evolutionary computation. Nature Machine Intelligence, 3(1), pp.9-15.Hansen, N. and Auger, A., 2014.Principled design of continuous stochastic search: From theory to practice. Theory and Principled Methods for the Design of Metaheuristics, pp.145-180.De Jong, K.A., 2006.Evolutionary computation: A unified approach. MIT Press.Beyer, H.G. and Deb, K., 2001.On self-adaptive features in real-parameter evolutionary algorithms. IEEE Transactions on Evolutionary Computation, 5(3), pp.250-270.Salomon, R., 1998.Evolutionary algorithms and gradient search: Similarities and differences. IEEE Transactions on Evolutionary Computation, 2(2), pp.45-55.Fogel, D.B., 1998.Evolutionary computation: The fossil record. IEEE Press.Wolpert, D.H. and Macready, W.G., 1997.No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation, 1(1), pp.67-82.Bäck, T. and Schwefel, H.P., 1993.An overview of evolutionary algorithms for parameter optimization. Evolutionary Computation, 1(1), pp.1-23.Forrest, S., 1993.Genetic algorithms: Principles of natural selection applied to computation. Science, 261(5123), pp.872-878.TaxonomyBenchmarking [benchmarking-network+iohprofiler]Kudela, J., 2022.A critical problem in benchmarking and analysis of evolutionary computation methods. Nature Machine Intelligence, 4(12), pp.1238-1245.Meunier, L., Rakotoarison, H., Wong, P.K., Roziere, B., Rapin, J., Teytaud, O., Moreau, A. and Doerr, C., 2022.Black-box optimization revisited: Improving algorithm selection wizards through massive benchmarking. IEEE Transactions on Evolutionary Computation, 26(3), pp.490-500.Hansen, N., Auger, A., Ros, R., Mersmann, O., Tušar, T. and Brockhoff, D., 2021.COCO: A platform for comparing continuous optimizers in a black-box setting. Optimization Methods and Software, 36(1), pp.114-144.Auger, A. and Hansen, N., 2021, July.Benchmarking: State-of-the-art and beyond. In Proceedings of Genetic and Evolutionary Computation Conference Companion (pp. 339-340). ACM.Varelas, K., El Hara, O.A., Brockhoff, D., Hansen, N., Nguyen, D.M., Tušar, T. and Auger, A., 2020.Benchmarking large-scale continuous optimizers: The bbob-largescale testbed, a COCO software guide and beyond. Applied Soft Computing, 97, p.106737.Hansen, N., Ros, R., Mauny, N., Schoenauer, M. and Auger, A., 2011.Impacts of invariance in search: When CMA-ES and PSO face ill-conditioned and non-separable problems. Applied Soft Computing, 11(8), pp.5755-5769.Moré, J.J. and Wild, S.M., 2009.Benchmarking derivative-free optimization algorithms. SIAM Journal on Optimization, 20(1), pp.172-191.Whitley, D., Rana, S., Dzubera, J. and Mathias, K.E., 1996.Evaluating evolutionary algorithms. Artificial Intelligence, 85(1-2), pp.245-276.Salomon, R., 1996.Re-evaluating genetic algorithm performance under coordinate rotation of benchmark functions. A survey of some theoretical and practical aspects of genetic algorithms. BioSystems, 39(3), pp.263-278.Fogel, D.B. and Beyer, H.G., 1995.A note on the empirical evaluation of intermediate recombination. Evolutionary Computation, 3(4), pp.491-495.Moré, J.J., Garbow, B.S. and Hillstrom, K.E., 1981.Testing unconstrained optimization software. ACM Transactions on Mathematical Software, 7(1), pp.17-41.Evolution Strategy (ES) [A visual guide to ES+[Li et al., 2020]+[Akimoto&Hansen, 2022, GECCO-Companion]]Akimoto, Y., Auger, A., Glasmachers, T. and Morinaga, D., 2022.Global linear convergence of evolution strategies on more than smooth strongly convex functions. SIAM Journal on Optimization, 32(2), pp.1402-1429.Glasmachers, T. and Krause, O., 2022.Convergence analysis of the Hessian estimation evolution strategy. ECJ, 30(1), pp.27-50.He, X., Zheng, Z. and Zhou, Y., 2021.MMES: Mixture model-based evolution strategy for large-scale optimization. IEEE-TEVC, 25(2), pp.320-333.Akimoto, Y. and Hansen, N., 2020.Diagonal acceleration for covariance matrix adaptation evolution strategies. ECJ, 28(3), pp.405-435.Beyer, H.G., 2020, July.Design principles for matrix adaptation evolution strategies. In ACM-GECCO Companion (pp. 682-700). ACM.Choromanski, K., Pacchiano, A., Parker-Holder, J. and Tang, Y., 2019.From complexity to simplicity: Adaptive es-active subspaces for blackbox optimization. In NeurIPS.Varelas, K., Auger, A., Brockhoff, D., Hansen, N., ElHara, O.A., Semet, Y., Kassab, R. and Barbaresco, F., 2018, September.A comparative study of large-scale variants of CMA-ES. In PPSN (pp. 3-15). Springer, Cham.Li, Z. and Zhang, Q., 2018.A simple yet efficient evolution strategy for large-scale black-box optimization. IEEE-TEVC, 22(5), pp.637-646.Lehman, J., Chen, J., Clune, J. and Stanley, K.O., 2018, July.ES is more than just a traditional finite-difference approximator. In ACM-GECCO (pp. 450-457). ACM.Ollivier, Y., Arnold, L., Auger, A. and Hansen, N., 2017.Information-geometric optimization algorithms: A unifying picture via invariance principles. Journal of Machine Learning Research, 18(18), pp.1-65.Loshchilov, I., 2017.LM-CMA: An alternative to L-BFGS for large-scale black box optimization. ECJ, 25(1), pp.143-171. [ Loshchilov, I., 2014, July.A computationally efficient limited memory CMA-ES for large scale optimization. In ACM-GECCO (pp. 397-404). ACM. ] + [ Loshchilov, I., Glasmachers, T. and Beyer, H.G., 2019.Large scale black-box optimization by limited-memory matrix adaptation. IEEE-TEVC, 23(2), pp.353-358. ]Beyer, H.G. and Sendhoff, B., 2017.Simplify your covariance matrix adaptation evolution strategy. IEEE Transactions on Evolutionary Computation, 21(5), pp.746-759.Krause, O., Arbonès, D.R. and Igel, C., 2016.CMA-ES with optimal covariance update and storage complexity. In Advances in Neural Information Processing Systems, 29, pp.370-378.Akimoto, Y. and Hansen, N., 2016, July.Projection-based restricted covariance matrix adaptation for high dimension. In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 197-204). ACM.Auger, A. and Hansen, N., 2016.Linear convergence of comparison-based step-size adaptive randomized search via stability of Markov chains. SIAM Journal on Optimization, 26(3), pp.1589-1624.Krause, O. and Igel, C., 2015, January.A more efficient rank-one covariance matrix update for evolution strategies. In Proceedings of ACM Conference on Foundations of Genetic Algorithms (pp. 129-136). ACM.Hansen, N., Arnold, D.V. and Auger, A., 2015.Evolution strategies. In Springer Handbook of Computational Intelligence (pp. 871-898). Springer, Berlin, Heidelberg.Diouane, Y., Gratton, S. and Vicente, L.N., 2015.Globally convergent evolution strategies. 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Journal of Optimization Theory and Applications, 66, pp.337-343. ] + [ Appel, M.J., Labarre, R. and Radulovic, D., 2004.On accelerated random search. SIAM Journal on Optimization, 14(3), pp.708-731. ]Solis, F.J. and Wets, R.J.B., 1981.Minimization by random search techniques. Mathematics of Operations Research, 6(1), pp.19-30.Schumer, M.A. and Steiglitz, K., 1968.Adaptive step size random search. IEEE Transactions on Automatic Control, 13(3), pp.270-276.Rastrigin, L.A., 1963.The convergence of the random search method in the extremal control of a many parameter system. Automaton & Remote Control, 24, pp.1337-1342. [ Rastrigin, L.A., 1986.Random search as a method for optimization and adaptation. In Stochastic Optimization. ]Brooks, S.H., 1958.A discussion of random methods for seeking maxima. Operations Research, 6(2), pp.244-251.Bayesian Optimization (BO)https://bayesoptbook.com/https://bayesopt-tutorial.github.io/Wang, L., Fonseca, R. and Tian, Y., 2020.Learning search space partition for black-box optimization using monte carlo tree search. Advances in Neural Information Processing Systems, 33, pp.19511-19522. [Python]Jones, D.R., Schonlau, M. and Welch, W.J., 1998.Efficient global optimization of expensive black-box functions. Journal of Global Optimization, 13(4), pp.455-492.Automated Machine Learning (AutoML)Hutter, F., Kotthoff, L. and Vanschoren, J., 2019.Automated machine learning: Methods, systems, challenges. Springer Nature.Hoos, H.H., 2011.Automated algorithm configuration and parameter tuning. In Autonomous Search (pp. 37-71). Springer, Berlin, Heidelberg.SoftwareCustódio, A.L., Scheinberg, K. and Nunes Vicente, L., 2017.Methodologies and software for derivative-free optimization. Advances and Trends in Optimization with Engineering Applications, pp.495-506.Rios, L.M. and Sahinidis, N.V., 2013.Derivative-free optimization: A review of algorithms and comparison of software implementations. Journal of Global Optimization, 56, pp.1247-1293.Schaul, T., Bayer, J., Wierstra, D., Sun, Y., Felder, M., Sehnke, F., Rückstieß, T. and Schmidhuber, J., 2010.PyBrain. Journal of Machine Learning Research, 11(24), pp.743-746.Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P., 2007.Numerical recipes: The art of scientific computing. Cambridge University Press. (See Chapter 10. Minimization or maximization of functions.)Sponsor for PyPop7This open-source Python library for black-box optimization is now supported by Shenzhen Fundamental Research Program under Grant No. JCYJ20200109141235597 (￥2,000,000 from 2021 to 2023), granted to Prof. Shi (CSE, SUSTech @ Shenzhen, China), and actively developed by three of his group members (e.g.,Q.Q. Duan,C. Shao,G.C. Zhou).Citation for PyPop7If this open-source pure-Python library is used in your paper or project, it is highly welcomed to cite the following arXivpreprintpaper:Duan, Q., Zhou, G., Shao, C., Wang, Z., Feng, M., Yang, Y., Zhao, Q. and Shi, Y., 2022. PyPop7: A pure-Python library for population-based black-box optimization. arXiv preprint arXiv:2212.05652.