pdfo.bobyqa(fun, x0, args=(), bounds=None, options=None)[source]#

Bounded Optimization BY Quadratic Approximations.

Deprecated since version 1.3: Calling the BOBYQA solver via the bobyqa function is deprecated. The BOBYQA solver remains available in PDFO. Call the pdfo function with the argument method='bobyqa' to use it.

fun: callable

Objective function to be minimized.

fun(x, *args) -> float

where x is an array with shape (n,) and args is a tuple.

x0: ndarray, shape (n,)

Initial guess.

args: tuple, optional

Parameters of the objective function. For example,

pdfo(fun, x0, args, ...)

is equivalent to

pdfo(lambda x: fun(x, *args), x0, ...)

bounds: {Bounds, ndarray, shape (n, 2)}, optional

Bound constraints of the problem. It can be one of the cases below.

  1. An instance of Bounds.

  2. An ndarray with shape (n, 2). The bound constraint for x[i] is bounds[i, 0] <= x[i] <= bounds[i, 1]. Set bounds[i, 0] to \(-\infty\) or None if there is no lower bound, and set bounds[i, 1] to \(\infty\) or None if there is no upper bound.

options: dict, optional

The options passed to the solver. It contains optionally:

rhobeg: float, optional

Initial value of the trust region radius, which should be a positive scalar. Typically, options['rhobeg'] should be in the order of one tenth of the greatest expected change to a variable. By default, it is min(1, min(ub - lb) / 4), and 0.5 if the problem is scaled.

rhoend: float, optional

Final value of the trust region radius, which should be a positive scalar. options['rhoend'] should indicate the accuracy required in the final values of the variables. Moreover, options['rhoend'] should be no more than options['rhobeg'] and is by default 1e-6.

maxfev: int, optional

Upper bound of the number of calls of the objective function fun. Its value must be not less than options['npt'] + 1. By default, it is 500 * n.

npt: int, optional

Number of interpolation points of each model used in Powell’s Fortran code.

ftarget: float, optional

Target value of the objective function. If a feasible iterate achieves an objective function value lower or equal to `options['ftarget'], the algorithm stops immediately. By default, it is \(-\infty\).

scale: bool, optional

Whether to scale the problem according to the bound constraints. By default, it is False. If the problem is to be scaled, then rhobeg and rhoend will be used as the initial and final trust region radii for the scaled problem.

honour_x0: bool, optional

Whether to respect the user-defined x0. By default, it is False.

quiet: bool, optional

Whether the interface is quiet. If it is set to True, the output message will not be printed. This flag does not interfere with the warning and error printing.

classical: bool, optional

Whether to call the classical Powell code or not. It is not encouraged in production. By default, it is False.

debug: bool, optional

Debugging flag. It is not encouraged in production. By default, it is False.

chkfunval: bool, optional

Flag used when debugging. If both options['debug'] and options['chkfunval'] are True, an extra function/constraint evaluation would be performed to check whether the returned values of objective function and constraint match the returned x. By default, it is False.

res: OptimizeResult

The results of the solver. Check OptimizeResult for a description of the attributes.

See also


Powell’s Derivative-Free Optimization solvers.


Unconstrained Optimization BY Quadratic Approximation.


NEW Unconstrained Optimization Algorithm.


LINearly Constrained Optimization Algorithm.


Constrained Optimization BY Linear Approximations.



M. J. D. Powell. The BOBYQA algorithm for bound constrained optimization without derivatives. Technical Report DAMTP 2009/NA06, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 2009.


The following example shows how to solve a simple bound-constrained optimization problem. The problem considered below should be solved with a derivative-based method. It is used here only as an illustration.

We consider the 2-dimensional problem

\[\begin{split}\min_{x, y \in \R} \quad x^2 + y^2 \quad \text{s.t.} \quad \left\{ \begin{array}{l} 0 \le x \le 2,\\ 1 / 2 \le y \le 3. \end{array} \right.\end{split}\]

We solve this problem using bobyqa starting from the initial guess \((x_0, y_0) = (0, 1)\) with at most 200 function evaluations.

>>> from pdfo import Bounds, bobyqa
>>> bounds = Bounds([0, 0.5], [2, 3])
>>> options = {'maxfev': 200}
>>> res = bobyqa(lambda x: x[0]**2 + x[1]**2, [0, 1], bounds=bounds, options=options)
>>> res.x
array([0. , 0.5])

Note that bobyqa can also be used to solve unconstrained problems.