Random Hill Climber (RHC)

class pypop7.optimizers.rs.rhc.RHC(problem, options)

Random (stochastic) Hill Climber (RHC).


Currently RHC only supports normally-distributed random sampling during optimization. It often suffers from slow convergence for large-scale black-box optimization (LSBBO), owing to its relatively limited exploration ability originating from its individual-based sampling strategy. Therefore, it is highly recommended to first attempt more advanced (e.g. population-based) methods for LSBBO.

“The hill-climbing search algorithm is the most basic local search technique. They have two key advantages: (1) they use very little memory; and (2) they can often find reasonable solutions in large or infinite state spaces for which systematic algorithms are unsuitable.”—[Russell&Norvig, 2021]

AKA “stochastic local search (steepest ascent or greedy search)”—[Murphy., 2022].

  • problem (dict) –

    problem arguments with the following common settings (keys):
    • ’fitness_function’ - objective function to be minimized (func),

    • ’ndim_problem’ - number of dimensionality (int),

    • ’upper_boundary’ - upper boundary of search range (array_like),

    • ’lower_boundary’ - lower boundary of search range (array_like).

  • options (dict) –

    optimizer options with the following common settings (keys):
    • ’max_function_evaluations’ - maximum of function evaluations (int, default: np.Inf),

    • ’max_runtime’ - maximal runtime to be allowed (float, default: np.Inf),

    • ’seed_rng’ - seed for random number generation needed to be explicitly set (int);

    and with the following particular settings (keys):
    • ’sigma’ - initial global step-size (float),

    • ’x’ - initial (starting) point (array_like),

      • if not given, it will draw a random sample from the uniform distribution whose search range is bounded by problem[‘lower_boundary’] and problem[‘upper_boundary’], when init_distribution is 1. Otherwise, standard normal distributed random sampling is used.

    • ’init_distribution’ - random sampling distribution for starting-point initialization (int, default: 1). Only when x is not set explicitly, it will be used.

      • 1: uniform distributed random sampling only for starting-point initialization,

      • 0: standard normal distributed random sampling only for starting-point initialization.


Use the optimizer to minimize the well-known test function Rosenbrock:

 1>>> import numpy
 2>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
 3>>> from pypop7.optimizers.rs.rhc import RHC
 4>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
 5...            'ndim_problem': 2,
 6...            'lower_boundary': -5*numpy.ones((2,)),
 7...            'upper_boundary': 5*numpy.ones((2,))}
 8>>> options = {'max_function_evaluations': 5000,  # set optimizer options
 9...            'seed_rng': 2022,
10...            'x': 3*numpy.ones((2,)),
11...            'sigma': 0.1}
12>>> rhc = RHC(problem, options)  # initialize the optimizer class
13>>> results = rhc.optimize()  # run the optimization process
14>>> # return the number of used function evaluations and found best-so-far fitness
15>>> print(f"RHC: {results['n_function_evaluations']}, {results['best_so_far_y']}")
16RHC: 5000, 7.13722829962456e-05

For its correctness checking of coding, refer to this code-based repeatability report for more details.


random sampling distribution for starting-point initialization.




global step-size (fixed during optimization).




initial (starting) point.




https://probml.github.io/pml-book/book2.html (See CHAPTER 6.7 Derivative-free optimization)

Russell, S. and Norvig P., 2021. Artificial intelligence: A modern approach (Global Edition). Pearson Education. http://aima.cs.berkeley.edu/ (See CHAPTER 4: SEARCH IN COMPLEX ENVIRONMENTS)

Hoos, H.H. and Stützle, T., 2004. Stochastic local search: Foundations and applications. Elsevier. https://www.elsevier.com/books/stochastic-local-search/hoos/978-1-55860-872-6