Gradient-based minimization of a non-expensive scalar function

Driver:

ScipyMinimizer

Download script: scipy_minimization.m

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,

\[ \begin{align}\begin{aligned}&\text{min.}\,& f(x_1,x_2) = 2\cdot10 + \sum_{i=1,2} \left(x_i^2 - 10\cos(2\pi x_i)\right)\\&\text{s.t.}\,& \sqrt{x_1^2 + x_2^2} \leq 1.5.\end{aligned}\end{align} \]
 1client = jcmoptimizer.Client(); 
 2
 3% Definition of the search domain
 4design_space = { ...
 5    struct('name', 'x1', 'type', 'continuous', 'domain', [-1.5,1.5]), ...
 6    struct('name', 'x2', 'type', 'continuous', 'domain', [-1.5,1.5]) ...
 7};
 8
 9% Definition of fixed environment parameter
10environment = {...     
11   struct('name', 'radius', 'type', 'fixed', 'domain', 1.5) ...
12};
13
14% Definition of a constraint on the search domain
15constraints = {...
16   struct('name', 'circle', 'expression', 'sqrt(x1^2 + x2^2) <= radius')...
17};
18
19 % Creation of the study object with study_id 'scipy_minimization'
20study = client.create_study( ...
21    'design_space', design_space, ...
22    'environment', environment, ...
23    'constraints', constraints,...
24    'driver','ScipyMinimizer',...
25    'study_name','Gradient-based minimization of a non-expensive scalar function',...
26    'study_id', 'scipy_minimization');
27
28study.configure('max_iter', 30, 'num_initial', 6, 'jac', true, 'method', 'SLSQP');
29
30% Evaluation of the black-box function for specified design parameters
31function observation = evaluate(study, sample)
32
33    observation = study.new_observation();
34    observation.add(10*2 ...
35                    + (sample.x1^2 - 10*cos(2*pi*sample.x1)) ...
36                    + (sample.x2^2 - 10*cos(2*pi*sample.x2)) ...
37                   );
38    observation.add(2*sample.x1 + 2*pi*sin(2*pi*sample.x1), ...
39                    'derivative', 'x1');
40    observation.add(2*sample.x2 + 2*pi*sin(2*pi*sample.x2), ...
41                    'derivative', 'x2');     
42    
43end  
44
45% Run the minimization
46study.set_evaluator(@evaluate);
47study.run(); 
48
49best = study.driver.best_sample;
50fprintf('Best sample at x1=%0.3e, x2=%0.3e\n', best.x1, best.x2) 
51
52client.shutdown_server();