scipy.optimize.

line_search#

scipy.optimize.line_search(f, myfprime, xk, pk, gfk=None, old_fval=None, old_old_fval=None, args=(), c1=0.0001, c2=0.9, amax=None, extra_condition=None, maxiter=10)[source]#

Find alpha that satisfies strong Wolfe conditions.

Parameters:

fcallable f(x,*args): Objective function.
myfprimecallable f’(x,*args): Objective function gradient.
xkndarray: Starting point.
pkndarray: Search direction. The search direction must be a descent direction for the algorithm to converge.
gfkndarray, optional: Gradient value for x=xk (xk being the current parameter estimate). Will be recomputed if omitted.
old_fvalfloat, optional: Function value for x=xk. Will be recomputed if omitted.
old_old_fvalfloat, optional: Function value for the point preceding x=xk.
argstuple, optional: Additional arguments passed to objective function.
c1float, optional: Parameter for Armijo condition rule.
c2float, optional: Parameter for curvature condition rule.
amaxfloat, optional: Maximum step size
extra_conditioncallable, optional: A callable of the form extra_condition(alpha, x, f, g) returning a boolean. Arguments are the proposed step alpha and the corresponding x, f and g values. The line search accepts the value of alpha only if this callable returns True. If the callable returns False for the step length, the algorithm will continue with new iterates. The callable is only called for iterates satisfying the strong Wolfe conditions.
maxiterint, optional: Maximum number of iterations to perform.

Returns:

alphafloat or None: Alpha for which x_new = x0 + alpha * pk, or None if the line search algorithm did not converge.
fcint: Number of function evaluations made.
gcint: Number of gradient evaluations made.
new_fvalfloat or None: New function value f(x_new)=f(x0+alpha*pk), or None if the line search algorithm did not converge.
old_fvalfloat: Old function value f(x0).
new_slopefloat or None: The local slope along the search direction at the new value <myfprime(x_new), pk>, or None if the line search algorithm did not converge.

Notes

Uses the line search algorithm to enforce strong Wolfe conditions. See Wright and Nocedal, ‘Numerical Optimization’, 1999, pp. 59-61.

The search direction pk must be a descent direction (e.g. -myfprime(xk)) to find a step length that satisfies the strong Wolfe conditions. If the search direction is not a descent direction (e.g. myfprime(xk)), then alpha, new_fval, and new_slope will be None.

Examples

>>> import numpy as np
>>> from scipy.optimize import line_search

A objective function and its gradient are defined.

>>> def obj_func(x):
...     return (x[0])**2+(x[1])**2
>>> def obj_grad(x):
...     return [2*x[0], 2*x[1]]

We can find alpha that satisfies strong Wolfe conditions.

>>> start_point = np.array([1.8, 1.7])
>>> search_gradient = np.array([-1.0, -1.0])
>>> line_search(obj_func, obj_grad, start_point, search_gradient)
(1.0, 2, 1, 1.1300000000000001, 6.13, [1.6, 1.4])