DX-NES-ICI [1] is a Natural Evolution Strategy (NES) for Mixed-Integer Black-Box Optimization (MI-BBO). DX-NES-ICI reportedly improves the performance of DX-NES-IC [2], one of the most promising continuous BBO methods, on MI-BBO problems. Simultaneously, DX-NES-ICI outperforms CMA-ES w. Margin [3], one of the most leading MI-BBO methods.

You need NumPy and SciPy that are the packages for scientific computing.

Please install via pip.

$ pip install dxnesici

Set the number of dimensions (dim), dimensions of continuous variables (dim_co), and dimensions of integer variables (dim_int). Then, set objective function and the domains of integer variables, where the decision variables must be arranged in such a way as to concatenate a (dim_co)-dimensional continuous vector and a (dim_int)-dimensional integer vector. Note that the number of elements of the domain in an integer variable must be greater than or equal to 2.

dim = 20
dim_co = dim // 2
dim_int = dim // 2
domain_int = [list(range(-10, 11)) for _ in range(dim_int)]
def n_int_tablet(x):
    xbar = np.array(x)
    xbar[dim_co:] = np.round(xbar[dim_co:])
    xbar[:dim_co] *= 100
    return np.sum(xbar**2)

Set initial mean vector (m), initial step size (sigma), and population size (lamb). Note that population size should be an even number. Set the minimum marginal probability (margin). 1.0 / (dim * lamb) [3] is the recommended value of minimum marginal probability.

m = np.ones([dim, 1]) * 2.
sigma = 1.0
lamb = 6
margin = 1.0 / (dim * lamb)

Pass variables to construct DXNESICI. You must pass the maximal number of evaluations and a target evaluation value to run the optimizer. Return values are a success flag, the best solution in the last generation, and the best evaluation value in the last generation.

dxnesici = DXNESICI(dim_co, domain_int, n_int_tablet, m, sigma, lamb, margin)
success, x_best, f_best = dxnesici.optimize(dim * 1e4, 1e-10)

• Version 1.0.3 (2023-12-09)
  • Update README.
• Version 1.0.2 (2023-12-09)
  • Fixed a bug in the return value of the best solution.
  • Fixed misleading implementations of functions in sample programs. Note that in these sample programs, benchmark functions return exactly the same values before and after this fix.
  • Add stdout of #Eval when DX-NES-ICI finish.
• Version 1.0.1 (2023-4-20)
  • First implementation.

1. Koki Ikeda and Isao Ono. 2023. Natural Evolution Strategy for Mixed-Integer Black-Box Optimization. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO ’23). 8 pages. https://doi.org/10.1145/3583131.3590518

2. Masahiro Nomura, Nobuyuki Sakai, Nobusumi Fukushima, and Isao Ono. 2021. Distance-weighted Exponential Natural Evolution Strategy for Implicitly Constrained Black-Box Function Optimization. In IEEE Congress on Evolutionary Computation (CEC ’21). 1099–1106. https://doi.org/10.1109/CEC45853.2021.9504865

3. Ryoki Hamano, Shota Saito, Masahiro Nomura, and Shinichi Shirakawa. 2022. CMA-ES with Margin: Lower-Bounding Marginal Probability for Mixed-Integer Black-Box Optimization. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO ’22). 639–647. https://doi.org/10.1145/3512290.3528827