Gradient-based minimization of a non-expensive scalar function

Driver:

ScipyMinimizer

Download script:

scipy_minimization.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 known derivatives. In this case a global optimization can be performed by a set of gradient-based local optimizations starting at different initial points. We start independent minimizations from six initial points (num_initial=6) and allow for two parallel evaluations of the objective function (num_parallel=2).

As an example, the 2D Rastrigin function on a circular domain is minimized,

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

import sys,os
import numpy as np
import time

from jcmoptimizer import Server, Client, Study, Obseravtion
server = Server()
client = Client(host=server.host)


# 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)},
]

# 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 'scipy_minimization'
study = client.create_study(
    design_space=design_space,
    environment=environment,
    constraints=constraints,
    driver="ScipyMinimizer",
    study_name="Gradient-based minimization of a non-expensive scalar function",
    study_id="scipy_minimization"
)

# Configure study parameters
study.configure(max_iter=80, num_parallel=2, num_initial=6, jac=True, method="SLSQP")

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

    observation = study.new_observation()
    observation.add(10*2
                + (x1**2-10*np.cos(2*np.pi*x1)) 
                + (x2**2-10*np.cos(2*np.pi*x2))
            )
    observation.add(2*x1+2*np.pi*np.sin(2*np.pi*x1),
                    derivative="x1")
    observation.add(2*x2+2*np.pi*np.sin(2*np.pi*x2),
                    derivative="x2")
    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}")