Global minimization of a non-expensive scalar function with discrete parameters

Driver:

CMAES

Download script:

cma_es.py

The target of the study is to minimize a scalar function. The scalar function is assumed to be inexpensive to evaluate (i.e. evaluation time shorter than a second) and to have no known derivatives. In this case a heuristic global optimization is advisable. CMA-ES is an evolutionary optimization algorithm that draws population members from a multivariate Gaussian distribution.

As an example, a 3D Rastrigin-like function on a mixed circular and discrete domain is minimized,

\[ \begin{align}\begin{aligned}&\text{min.}\,& f(x_1,x_2,x_3) = 3\cdot10 + \sum_{i=1}^3 \left(x_i^2 - 10\cos(2\pi x_i)\right)\\&\text{s.t.}\,& \sqrt{x_1^2 + x_2^2} \leq 1.5 \wedge x_3 \in \{-1,0,1\}.\end{aligned}\end{align} \]

import sys,os
import numpy as np
import time

from jcmoptimizer import Client, Study, Obseravtion
client = Client()


# Definition of the search domain
design_space = [
    {'name': 'x1', 'type': 'continuous', 'domain': (-1.5,1.5)}, 
    {'name': 'x2', 'type': 'continuous', 'domain': (-1.5,1.5)},
    {'name': 'x3', 'type': 'discrete', 'domain': (1.0,0.0,1.0)},
]

# Definition of fixed environment parameter
environment = [
    {'name': 'radius', 'type': 'fixed', 'domain': 1.5},
]

# Definition of a constraint on the search domain
constraints = [
    {'name': 'circle', 'expression': 'sqrt(x1^2 + x2^2) <= radius'}
]

# Creation of the study object with study_id 'cma_es'
study = client.create_study(
    design_space=design_space,
    environment=environment,
    constraints=constraints,
    driver="CMAES",
    study_name="Global minimization of a non-expensive scalar function with discrete parameters",
    study_id="cma_es"
)
# Configure study parameters
study.configure(max_iter=80, num_parallel=2)

# Evaluation of the black-box function for specified design parameters
def evaluate(study: Study, x1: float, x2: float, x3: float, radius: float) -> Observation:

    observation = study.new_observation()
    observation.add(10*3
                + (x1**2-10*np.cos(2*np.pi*x1)) 
                + (x2**2-10*np.cos(2*np.pi*x2))
                + (x3**2-10*np.cos(2*np.pi*x3))
            )return observation

# Run the minimization
study.set_evaluator(evaluate)
study.run()
best = study.driver.best_sample
print(f"Best sample at: x1={best['x1']:.3f}, x2={best['x2']:.3f}, x3={best['x3']:.1f}")

client.shutdown_server()