L_BFGS_B_Optimization¶

Purpose¶

The purpose of the driver is to identify a parameter vector $\mathbf{p}\in\mathcal{X}\subset\mathbb{R}^d$ that minimizes the value of an objective function $f_\text{objective}: \mathcal{X} \rightarrow \mathbb{R}$ . The search domain $\mathcal{X}$ is bounded by box constraints $l_i\leq p_i \leq u_i$ for $1\leq i\leq d$ and may be subject to several constraints $c_j: \mathbb{R}^d \rightarrow \mathbb{R}$ such that $\mathbf{p} \in \mathcal{X}$ only if $c_j(\mathbf{p}) \leq 0$ (see jcmwave_optimizer_create_study()).

The driver uses the L-BFGS-B algorithm to perform a gradient-based minimization. We recommend to use the driver if an exact convergence towards a local or global minimum is required. If no derivative information are available, a convergence can be better performed with the derivative-free downhill-simplex minimization.

The implementation of the driver is based on the open source implementation of scipy (see https://docs.scipy.org/doc/scipy/reference/optimize.minimize-lbfgsb.html). It is extended to support constraints and a parallel optimization by starting several independent minimizers at different positions.

Usage Example¶

addpath(fullfile(getenv('JCMROOT'), 'ThirdPartySupport', 'Matlab'));
client = jcmwave_optimizer_client();

% Definition of the search domain
domain = {...
    struct('name','x1', 'type','continuous', 'domain', [-1.5,1.5]),...
    struct('name','x2', 'type','continuous', 'domain', [-1.5,1.5]),...
    struct('name','radius', 'type','fixed', 'domain', 2)...
};

% Definition of a constraint on the search domain
constraints = [...
    struct('name', 'circle', 'constraint','sqrt(x1^2 + x2^2) - radius')...
];

% Creation of the study object with study_id 'example'
study = client.create_study('domain',domain, 'constraints',constraints, ...
                'driver','L_BFGS_B_Optimization',...
                'name','L_BFGS_B_Optimization example', ...
                'study_id','L_BFGS_B_Optimization_example');

% Definition of a simple analytic objective function.
% Typically, the objective value is derived from a FEM simulation
% using jcmwave.solve(...)
function observation = objective(sample)
  pause(2.0); % makes objective expensive
  observation = study.new_observation();

  x1 = sample.x1;
  x2 = sample.x2;
  observation.add(10*2 + (x1.^2-10*cos(2*pi*x1)) + (x2.^2-10*cos(2*pi*x2)));

  %derivative w.r.t. x1
  observation.add(2*x1 + 20*pi*sin(2*pi*x1), 'x1');
  %derivative w.r.t. x2
  observation.add(2*x2 + 20*pi*sin(2*pi*x2), 'x2');
  end

% Set study parameters
study.set_parameters('max_iter',25, 'num_initial',3,...
                     'jac',true, 'initial_samples', [[0.5,0.5];[-0.5,-0.5]]);


% Run the minimization
while(not(study.is_done))
    sug = study.get_suggestion();
    obs = objective(sug.sample);
    study.add_observation(obs, sug.id);
end

info = study.info();
fprintf('\nMinimum %0.3e found at (x1=%0.3e, x2=%0.3e)',...
        info.min_objective, info.min_params.x1, info.min_params.x2)

Parameters¶

The following parameters can be set by calling, e.g.

study.set_parameters('example_parameter1',[1,2,3], 'example_parameter2',true);

max_iter (int):	Maximum number of evaluations of the objective function (default: inf)

max_time (int):	Maximum run time in seconds (default: inf)

num_parallel (int):
	Number of parallel observations of the objective function (default: 1)

eps (float):	Stopping criterium. Minimum distance in the parameter space to the currently known minimum (default: 0.0)

min_val (float):
	Stopping criterium. Minimum value of the objective function (default: -inf)

num_initial (int):
	Number of independent initial optimizers (default: 1)

max_num_minimizers (int):
	If a minimizer has converged, it is restarted at another position. If max_num_minimizers threads have converged, the optimization is stopped (default: inf)

sobol_sequence (bool):
	If true, all initial samples are taken from a Sobol sequence. This typically improves the coverage of the parameter space. (default: True)

jac (bool):	If true, the gradient is used for optimization (default: False)

step_size (float):
	Step size used for numerical approximation of the gradient (default: 1e-06)

f_tol (float):	The iteration stops when $(f^k - f^{k+1})/max\{\|f^k\|,\|f^{k+1}\|,1\} \leq ftol$ . (default: 2.2e-09)