pypop7 0.0.67

pypop7 (Pure-PYthon library of POPulation-based OPtimization)

PyPop7 (Pure-PYthon library of POPulation-based black-box OPtimization)

PyPop7 is a Pure-PYthon library of POPulation-based OPtimization for single-objective, real-parameter, black-box problems (currently actively maintained). Its main goal is to provide a unified interface and elegant implementations for Black-Box Optimization (BBO), particularly population-based optimizers, in order to facilitate research repeatability and possible real-world applications.

More specifically, for alleviating their notorious curse of dimensionality, the primary focus of PyPop7 is to cover their State Of The Art (SOTA) for Large-Scale Optimization (LSO), though many of their small/medium-scaled versions and variants are also included here (mainly for benchmarking purpose).

How to Use PyPop7

The following three simple steps are enough to utilize the optimization power of PyPop7:

1. Use pip to install pypop7 on the Python3-based virtual environment via venv or conda:

$ pip install pypop7

For simplicity, all required dependencies (except special cases) are automatically installed according to setup.cfg.

2. Define the objective/cost function (called fitness function in this library) for the optimization problem at hand,

3. Run one or more black-box optimizers on this optimization problem:

import numpy as np  # for numerical computation, which is also the computing engine of pypop7

# 2. Define your own objective function for the optimization problem at hand:
#   the below example is Rosenbrock, the notorious test function in the optimization community
def rosenbrock(x):
    return 100*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 settings
ndim_problem = 1000
problem = {'fitness_function': rosenbrock,  # cost function
           'ndim_problem': ndim_problem,  # dimension
           'lower_boundary': -5*np.ones((ndim_problem,)),  # search boundary
           'upper_boundary': 5*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.html
from pypop7.optimizers.es.lmmaes import LMMAES
# 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': 0.3,  # initial global step-size of search distribution (not necessarily optimal)
           'verbose': 500}
lmmaes = LMMAES(problem, options)  # initialize the optimizer
results = lmmaes.optimize()  # run its (time-consuming) search process
print(results)

Black-Box Optimizers (BBO)

Note that Ant Colony Optimization (ACO) and Tabu Search (TS) are not covered in this library, since they works mainly in the discrete search space. Furthermore, brute-force search (exhaustive/grid search) is also excluded in this library, since it works only for very low dimensions. In the future version, we will consider to add Simultaneous Perturbation Stochastic Approximation into this library.

---

• : indicates the specific BBO version for LSO (e.g., dimension >= 1000).
• : indicates the competitive (or de facto) BBO version for relatively low-dimensional problems (though it may work well under certain LSO circumstances).
• : indicates the baseline BBO version mainly for theoretical interest.

Note that this classification based only on dimensions 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 advanced Algorithm Selection techniques.

---

• 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]
  • 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 Philosophy

Given 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 for Beauty (Elegance)

  From the problem-solving perspective, 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 often unknown in advance (when without a prior knowledge). 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 the academic research perspective, however, we prefer so-called beautiful optimizers, though always keeping the No Free Lunch Theorems in mind. Typically, the beauty of one optimizer comes from the following attractive features: model novelty, competitive performance on at least one class of problems, theoretical insights (e.g., convergence), clarity/simplicity for understanding and implementation, and repeatability.

  If you find any to meet the above standard, welcome to launch issues or pulls. We will consider it to be included in the pypop7 library as soon as possible, if possible. Note that any superficial imitation to well-established optimizers (i.e. Old Wine in a New Bottle will be NOT considered here. Sometimes, several very complex optimizers could obtain the top rank on some competitions consisting of only artificially-constructed benchmark functions. However, these optimizers may become over-skilled on these artifacts. In our opinions, a good optimizer should be elegant and generalizable. If there is no persuasive application reported for it, we will not consider any very complex optimizer in this library, in order to aovid the possible repeatability and overfitting issues.

• Respect for Diversity

  Given the universality of black-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 share more or less similarities. On the other hand, they may also show significant differences (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 learning evolutionary computation and 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 for Originality

  For each optimizer included in PyPop7, 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 for Repeatability

  For randomized search, properly controlling randomness is very crucial to repeat numerical experiments. Here we follow the Random Sampling suggestions from NumPy. In other worlds, you must explicitly set the random seed for each optimizer. For more discussions about repeatability from 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 Efficiency

For LSO, computational efficiency is an indispensable performance criterion of DFO in the post-Moore era. To obtain high-performance computation as much as possible, NumPy is heavily used in this library as the base of numerical computation along with SciPy. Sometimes, Numba is also utilized, in order to further accelerate the wall-clock time.

References

For each algorithm family, we provide several representative applications published on some top-tier journals 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)
  • 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.
  • Taxonomy
• Benchmarking [ benchmarking-network ]
  • 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 evolution strategies ]
  • Glasmachers, T. and Krause, O., 2022. Convergence analysis of the Hessian estimation evolution strategy. Evolutionary Computation, 30(1), pp.27-50.
  • Akimoto, Y. and Hansen, N., 2022, July. CMA-ES and advanced adaptation mechanisms. In Proceedings of Annual Conference on Genetic and Evolutionary Computation Companion. ACM.
  • He, X., Zheng, Z. and Zhou, Y., 2021. MMES: Mixture model-based evolution strategy for large-scale optimization. IEEE Transactions on Evolutionary Computation, 25(2), pp.320-333.
  • Li, Z., Lin, X., Zhang, Q. and Liu, H., 2020. Evolution strategies for continuous optimization: A survey of the state-of-the-art. Swarm and Evolutionary Computation, 56, p.100694.
  • Akimoto, Y. and Hansen, N., 2020. Diagonal acceleration for covariance matrix adaptation evolution strategies. Evolutionary Computation, 28(3), pp.405-435.
  • Beyer, H.G., 2020, July. Design principles for matrix adaptation evolution strategies. In Proceedings of Annual Conference on Genetic and Evolutionary Computation 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 Advances in Neural Information Processing Systems.
  • Loshchilov, I., Glasmachers, T. and Beyer, H.G., 2019. Large scale black-box optimization by limited-memory matrix adaptation. IEEE Transactions on Evolutionary Computation, 23(2), pp.353-358.
  • Li, Z., Zhang, Q., Lin, X. and Zhen, H.L., 2018. Fast covariance matrix adaptation for large-scale black-box optimization. IEEE Transactions on Cybernetics, 50(5), pp.2073-2083.
  • 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 International Conference on Parallel Problem Solving from Nature (pp. 3-15). Springer, Cham.
  • Li, Z. and Zhang, Q., 2018. A simple yet efficient evolution strategy for large-scale black-box optimization. IEEE Transactions on Evolutionary Computation, 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 Proceedings of Annual Conference on Genetic and Evolutionary Computation (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. Evolutionary Computation, 25(1), pp.143-171.
  • 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.
  • 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.
  • Loshchilov, I., 2014, July. A computationally efficient limited memory CMA-ES for large scale optimization. In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 397-404). ACM.
  • Hansen, N., Atamna, A. and Auger, A., 2014, September. How to assess step-size adaptation mechanisms in randomised search. In International Conference on Parallel Problem Solving from Nature (pp. 60-69). Springer, Cham.
  • Akimoto, Y., Auger, A. and Hansen, N., 2014, July. Comparison-based natural gradient optimization in high dimension. In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 373-380). ACM.
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  • Applications: e.g., Tjanaka et al., 2023; Sun et al., 2022, ICML; Tjanaka et al., 2022, GECCO; Wang&Ponce, 2022, GECCO, Tian&Ha, 2022, EvoStar; Hansel et al., 2021; Anand et al., 2021, MLST; Nomura et al., 2021, AAAI; Liu et al., 2019, AAAI; Dong et al., 2019, CVPR; Ha&Schmidhuber, 2018, NeurIPS; Müller&Glasmachers, 2018, PPSN; Chrabąszcz et al., 2018, IJCAI; OpenAI, 2017; Zhang et al., 2017, Science; Heidrich-Meisner&Igel, 2009, ICML; Heidrich-Meisner&Igel, 2008, PPSN; Lipson&Pollack, 2000, Nature.
• Natural Evolution Strategies (NES)
  • Wierstra, D., Schaul, T., Glasmachers, T., Sun, Y., Peters, J. and Schmidhuber, J., 2014. Natural evolution strategies. Journal of Machine Learning Research, 15(1), pp.949-980.
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  • Wierstra, D., Schaul, T., Peters, J. and Schmidhuber, J., 2008, June. Natural evolution strategies. In IEEE Congress on Evolutionary Computation (pp. 3381-3387). IEEE.
  • Applications: e.g., Yu et al., USENIX Security; Flageat et al., 2023; Yan et al., 2023; Feng et al., 2023; Wei et al., 2022, IJCV; Agarwal et al., 2022, ICRA; Farid et al., 2022, CoRL; Feng et al., 2022, CVPR; Berliner et al., 2022, ICLR; Kirsch et al., 2022, AAAI; Jain et al., 2022, USENIX Security; Ilyas et al., 2018, ICML.
• Estimation of Distribution Algorithm (EDA)
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• Cross-Entropy Method (CEM)
  • Pinneri, C., Sawant, S., Blaes, S., Achterhold, J., Stueckler, J., Rolinek, M. and Martius, G., 2021, October. Sample-efficient cross-entropy method for real-time planning. In Conference on Robot Learning (pp. 1049-1065). PMLR.
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  • Applications: e.g., Wang&Ba,2020, ICLR; Hafner et al., 2019, ICML; Pourchot&Sigaud, 2019, ICLR; Simmons-Edler et al., 2019, ICML-RL4RealLife; Chua et al., 2018, NeurIPS; Duan et al., 2016, ICML; Kobilarov, 2012, IJRR.
• Differential Evolution (DE)
  • Storn, R.M. and Price, K.V. 1997. Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4), pp.341–359.
  • Applications: e.g., Mehta et al., 2021, JCAP; An et al., 2020, PNAS; Su et al., 2019, TEVC; Laganowsky et al., 2014, Nature.
• Particle Swarm Optimizer (PSO) [ swarm intelligence | scholarpedia ]
  • Sünnen, P., 2023. Analysis of a consensus-based optimization method on hypersurfaces and applications. Doctoral dissertation, Technische Universität München.
  • Fornasier, M., Huang, H., Pareschi, L. and Sünnen, P., 2021. Consensus-based optimization on the sphere: Convergence to global minimizers and machine learning. Journal of Machine Learning Research, 22(1), pp.10722-10776.
  • Carrillo, J.A., Choi, Y.P., Totzeck, C. and Tse, O., 2018. An analytical framework for consensus-based global optimization method. Mathematical Models and Methods in Applied Sciences, 28(06), pp.1037-1066.
  • Blackwell, T. and Kennedy, J., 2018. Impact of communication topology in particle swarm optimization. IEEE Transactions on Evolutionary Computation, 23(4), pp.689-702.
  • Pinnau, R., Totzeck, C., Tse, O. and Martin, S., 2017. A consensus-based model for global optimization and its mean-field limit. Mathematical Models and Methods in Applied Sciences, 27(01), pp.183-204.
  • Bonyadi, M.R. and Michalewicz, Z., 2017. Particle swarm optimization for single objective continuous space problems: A review. Evolutionary Computation, 25(1), pp.1-54.
  • Escalante, H.J., Montes, M. and Sucar, L.E., 2009. Particle swarm model selection. Journal of Machine Learning Research, 10(15), pp.405−440.
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  • Parsopoulos, K.E. and Vrahatis, M.N., 2002. Recent approaches to global optimization problems through particle swarm optimization. Natural Computing, 1(2), pp.235-306.
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  • Eberhart, R.C., Shi, Y. and Kennedy, J., 2001. Swarm intelligence. Elsevier.
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  • Eberhart, R. and Kennedy, J., 1995, October. A new optimizer using particle swarm theory. In Proceedings of International Symposium on Micro Machine and Human Science (pp. 39-43). IEEE.
  • Applications: e.g., [Reddy et al., 2023, TC; Zhang et al., 2022, CVPR; Yang et al., PRL, 2022; Guan et al., 2022, PRL; Zhong et al., 2022, CVPR; Singh&Hecke, 2021, PRL; Weiel, et al., 2021, Nature Mach. Intell; Wintermantel et al., 2020, PRL; Tang et al., 2019, TPAMI; Sheng et al., 2019, TPAMI; CMS Collaboration, 2019, JHEP; Wang et al., 2019, TVCG; Zhang et al., 2018, PRL; Leditzky et al., 2018, PRL; Pham et al., 2018, TPAMI; Villeneuve et al., 2017, Science; Choi et al., 2017, PRL; González-Echevarría, et al., 2017, TCAD; Zhu et al., 2017, PRL; Choi et al., 2017, ICCV; Pickup et al., 2016, IJCV; Li et al., 2015, PRL; Sharp et al., 2015, CHI; Taneja et al., 2015, TPAMI; Zhang et al., 2015, IJCV; Meyer et al., 2015, ICCV; Tompson et al., 2014, TOG; Baca et al., 2013, Cell; Li et al., PRL, 2013; Kawakami et al., 2013, IJCV; Kim et al., 2012, Nature; Rahmat-Samii et al., 2012, PIEEE; Oikonomidis et al., 2012, CVPR; Li et al., 2011, TPAMI; Zhao et al., 2011, PRL; Zhu et al., 2011, PRL; Lv et al., 2011, PRL; Hentschel&Sanders, 2010, PRL; Pontani&Conway, 2010, JGCD; Zhang et al., 2008, CVPR; Liebelt&Schertler, 2007, CVPR; Hassan et al., 2005, AIAA].
• Cooperative Coevolution (CC)
  • Gomez, F., Schmidhuber, J. and Miikkulainen, R., 2008. Accelerated neural evolution through cooperatively coevolved synapses. Journal of Machine Learning Research, 9(31), pp.937-965.
  • Panait, L., Tuyls, K. and Luke, S., 2008. Theoretical advantages of lenient learners: An evolutionary game theoretic perspective. Journal of Machine Learning Research, 9, pp.423-457.
  • Schmidhuber, J., Wierstra, D., Gagliolo, M. and Gomez, F., 2007. Training recurrent networks by evolino. Neural Computation, 19(3), pp.757-779.
  • Gomez, F.J. and Schmidhuber, J., 2005, June. Co-evolving recurrent neurons learn deep memory POMDPs. In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 491-498).
  • Fan, J., Lau, R. and Miikkulainen, R., 2003. Utilizing domain knowledge in neuroevolution. In International Conference on Machine Learning (pp. 170-177).
  • Potter, M.A. and De Jong, K.A., 2000. Cooperative coevolution: An architecture for evolving coadapted subcomponents. Evolutionary Computation, 8(1), pp.1-29.
  • Gomez, F.J. and Miikkulainen, R., 1999, July. Solving non-Markovian control tasks with neuroevolution. In Proceedings of International Joint Conference on Artificial Intelligence (pp. 1356-1361).
  • Moriarty, D.E. and Mikkulainen, R., 1996. Efficient reinforcement learning through symbiotic evolution. Machine Learning, 22(1), pp.11-32.
  • Moriarty, D.E. and Miikkulainen, R., 1995. Efficient learning from delayed rewards through symbiotic evolution. In International Conference on Machine Learning (pp. 396-404). Morgan Kaufmann.
  • Potter, M.A. and De Jong, K.A., 1994, October. A cooperative coevolutionary approach to function optimization. In International Conference on Parallel Problem Solving from Nature (pp. 249-257). Springer, Berlin, Heidelberg.
• Simultaneous Perturbation Stochastic Approximation (SPSA) [ https://www.jhuapl.edu/SPSA/ ]
• Simulated Annealing (SA)
  • Bouttier, C. and Gavra, I., 2019. Convergence rate of a simulated annealing algorithm with noisy observations. Journal of Machine Learning Research, 20(1), pp.127-171.
  • Gerber, M. and Bornn, L., 2017. Improving simulated annealing through derandomization. Journal of Global Optimization, 68(1), pp.189-217.
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  • Corana, A., Marchesi, M., Martini, C. and Ridella, S., 1987. Minimizing multimodal functions of continuous variables with the “simulated annealing” algorithm. ACM Transactions on Mathematical Software, 13(3), pp.262-280. [ Corrigenda ]
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  • Applications: e.g., Kolesov et al., 2016, IEEE-TPAMI
• Genetic Algorithm (GA)
  • Whitley, D., 2019. Next generation genetic algorithms: A user’s guide and tutorial. In Handbook of Metaheuristics (pp. 245-274). Springer, Cham.
  • Levine, D., 1997. Commentary—Genetic algorithms: A practitioner's view. INFORMS Journal on Computing, 9(3), pp.256-259.
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  • Forrest, S., 1993. Genetic algorithms: Principles of natural selection applied to computation. Science, 261(5123), pp.872-878.
  • De Jong, K.A., 1993. Are genetic algorithms function optimizer?. Foundations of Genetic Algorithms, pp.5-17.
  • Mitchell, M., Holland, J. and Forrest, S., 1993. When will a genetic algorithm outperform hill climbing. Advances in Neural Information Processing Systems (pp. 51-58).
  • Holland, J.H., 1992. Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence. MIT Press.
  • Holland, J.H., 1992. Genetic algorithms. Scientific American, 267(1), pp.66-73.
  • Whitley, D., 1989, December. The GENITOR algorithm and selection pressure: Why rank-based allocation of reproductive trials is best. In Proceedings of International Conference on Genetic Algorithms (pp. 116-121).
  • Goldberg, D.E. and Holland, J.H., 1988. Genetic algorithms and machine learning. Machine Learning, 3(2), pp.95-99.
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  • Applications: e.g., Wang, 2023, Harvard Ph.D. Dissertation; Lee et al., 2022, Science Robotics; Chen et al., 2020, Nature; Whitley et al., 1993, MLJ.
• Evolutionary Programming (EP)
  • Yao, X., Liu, Y. and Lin, G., 1999. Evolutionary programming made faster. IEEE Transactions on Evolutionary Computation, 3(2), pp.82-102.
  • Fogel, D.B., 1999. An overview of evolutionary programming. In Evolutionary Algorithms (pp. 89-109). Springer, New York, NY.
  • Fogel, D.B. and Fogel, L.J., 1995, September. An introduction to evolutionary programming. In European Conference on Artificial Evolution (pp. 21-33). Springer, Berlin, Heidelberg.
  • Fogel, D.B., 1994. Evolutionary programming: An introduction and some current directions. Statistics and Computing, 4(2), pp.113-129.
  • Bäck, T. and Schwefel, H.P., 1993. An overview of evolutionary algorithms for parameter optimization. Evolutionary Computation, 1(1), pp.1-23.
  • Applications: e.g., Hoorfar, 2007, IEEE-TAP; Cui et al., 2006, MS; Damavandi&Safavi-Naeini, 2005, IEEE-TCSI.
• Pattern Search
  • Audet, C., Le Digabel, S., Rochon Montplaisir, V. and Tribes, C., 2022. Algorithm XXXX: NOMAD version 4: Nonlinear optimization with the MADS algorithm. ACM Transactions on Mathematical Software.
  • Brent, R.P., 2013. Algorithms for minimization without derivatives. Courier Corporation.
  • Singer, S. and Nelder, J., 2009. Nelder-mead algorithm. Scholarpedia, 4(7), p.2928.
  • Kolda, T.G., Lewis, R.M. and Torczon, V., 2003. Optimization by direct search: New perspectives on some classical and modern methods. SIAM Review, 45(3), pp.385-482.
  • Lagarias, J.C., Reeds, J.A., Wright, M.H. and Wright, P.E., 1998. Convergence properties of the Nelder--Mead simplex method in low dimensions. SIAM Journal on Optimization, 9(1), pp.112-147.
  • Powell, M.J., 1998. Direct search algorithms for optimization calculations. Acta Numerica, 7, pp.287-336.
  • Torczon, V., 1997. On the convergence of pattern search algorithms. SIAM Journal on Optimization, 7(1), pp.1-25.
  • Barton, R.R. and Ivey Jr, J.S., 1996. Nelder-Mead simplex modifications for simulation optimization. Management Science, 42(7), pp.954-973.
  • Wright, M.H., 1996. Direct search methods: Once scorned, now respectable. Pitman Research Notes in Mathematics Series, pp.191-208.
  • Jones, D.R., Perttunen, C.D. and Stuckman, B.E., 1993. Lipschitzian optimization without the Lipschitz constant. Journal of Optimization Theory and Applications, 79(1), pp.157-181.
  • Nelder, J.A. and Mead, R., 1965. A simplex method for function minimization. The Computer Journal, 7(4), pp.308-313.
  • Powell, M.J., 1964. An efficient method for finding the minimum of a function of several variables without calculating derivatives. The Computer Journal, 7(2), pp.155-162.
  • Kaupe Jr, A.F., 1963. Algorithm 178: Direct search. Communications of the ACM, 6(6), pp.313-314.
  • Spang, III, H.A., 1962. A review of minimization techniques for nonlinear functions. SIAM Review, 4(4), pp.343-365.
  • Hooke, R. and Jeeves, T.A., 1961. “Direct search” solution of numerical and statistical problems. Journal of the ACM, 8(2), pp.212-229. [ Python - pymoo | This Week's Citation Classic ]
  • Box, G.E., 1957. Evolutionary operation: A method for increasing industrial productivity. Journal of the Royal Statistical Society: Series C (Applied Statistics), 6(2), pp.81-101.
  • Fermi, E. and Metropolis N., 1952. Numerical solution of a minimum problem. Los Alamos Scientific Lab., Los Alamos, NM.
    • Anderson, H.L., Davidon, W.C., Glicksman, M. and Kruse, U.E., 1955. Scattering of positive pions by hydrogen at 189 MeV. Physical Review, 100(1), p.279.
  • Applications: e.g., [NM: Gokhale et al., 2023, PNAS; Hayashi, 2022, Bernoulli; Vanunu et al., 2021, PNAS; Williams et al., 2021, PNAS; Fleishman et al., Science, 2020; Nanni et al., 2020, PNAS; Steinrücken et al., 2019, PNAS; Omran et al., 2019, Science; Sparrow et al., 2018, Nature; Prochazka&Vogl, 2017, PNAS; Murphy&Brincke, 2017, MS; Gillon et al., 2017, Nature; Aghaeepour et al., 2017, Science Immunol.; Sayegh et al., 2017, TS; Landis&Schraiber, 2017, PNAS; Kim et al., 2016, PNAS; Chan et al., 2014, MS; Chan et al., 2014, MKSC; Bajikar et al., 2014, PNAS; Lee et al., 2014, PNAS; Wang et al., 2012, PRL; Lau&Rubinstein, Nature, 2012; Brown et al., 2012, MS; Contreras et al., 2012, PNAS; Morlon et al., 2011, PNAS; Forstmann et al., 2010, PNAS; Balachander et al., 2009, MS; Jayanthi et al., 2009, MS; Farrell et al., 2009, PNAS; Forstmann et al., 2008, PNAS; Rouder et al., 2008, PNAS; Sapir et al., 2005, PNAS; Amonlirdviman et al., 2005, Science; Cowan et al., 2004, PS; Zhou et al., 2004, PS; Draganska&Jain, 2004, MS; Fain&Levitt, 2003, PNAS; Dennis et al., 2002, PNAS; Sudhir, 2001, MKSC; Rouder, 2001, PS; Wolszczan, 1994, Science; Polvani et al., 1990, Science; Lee et al., 1987, PNAS; Sabath et al., 1986, PNAS; Burch et al., 1985, PNAS; Regoeczi et al., 1982, PNAS; Brasseur et al., 1982, PNAS; Korn et al., 1981, Science; Dean et al., 1975, Science]; [HJ: Khaledian et al., 2018, IEEE-TMTT; Luhar et al., 2015, JFM; Paxton et al., 2013, ApJS; Schneider&Excoffier, 1999, Genetics; Ditchfield et al., 1971, JCP].
• Random Search (RS)
  • Bergstra, J. and Bengio, Y., 2012. Random search for hyper-parameter optimization. Journal of Machine Learning Research, 13(2).
  • Appel, M.J., Labarre, R. and Radulovic, D., 2004. On accelerated random search. SIAM Journal on Optimization, 14(3), pp.708-731.
  • Schmidhuber, J., Hochreiter, S. and Bengio, Y., 2001. Evaluating benchmark problems by random guessing. A Field Guide to Dynamical Recurrent Networks, pp.231-235.
  • Rosenstein, M.T. and Barto, A.G., 2001, August. Robot weightlifting by direct policy search. In International Joint Conference on Artificial Intelligence (pp. 839-846).
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  • Rastrigin, L.A., 1986. Random search as a method for optimization and adaptation. In Stochastic Optimization.
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  • 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.
  • 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.
• Software
  • Custódio, 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.
  • 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.)

Research Support

This 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. Yuhui Shi (CSE, SUSTech @ Shenzhen, China), and actively developed by three of his group members (e.g., Qiqi Duan, Chang Shao, Guochen Zhou).

Citation for PyPop7

If you use this library in your paper or project, it is highly welcomed to cite the following paper: 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.