proximal_gradient¶

odl.solvers.nonsmooth.proximal_gradient_solvers.proximal_gradient(x, f, g, gamma, niter, callback=None, **kwargs)[source]¶

(Accelerated) proximal gradient algorithm for convex optimization.

Also known as "Iterative Soft-Thresholding Algorithm" (ISTA). See [Beck2009] for more information.

This solver solves the convex optimization problem:

min_{x in X} f(x) + g(x)

where the proximal operator of f is known and g is differentiable.

Parameters:

xf.domain element: Starting point of the iteration, updated in-place.
fFunctional: The function f in the problem definition. Needs to have f.proximal.
gFunctional: The function g in the problem definition. Needs to have g.gradient.
gammapositive float: Step size parameter.
niternon-negative int, optional: Number of iterations.
callbackcallable, optional: Function called with the current iterate after each iteration.

Other Parameters:

lamfloat or callable, optional: Overrelaxation step size. If callable, it should take an index (starting at zero) and return the corresponding step size. Default: 1.0

Notes

The problem of interest is

$\min_{x \in X} f(x) + g(x),$

where the formal conditions are that $f : X \to \mathbb{R}$ is proper, convex and lower-semicontinuous, and $g : X \to \mathbb{R}$ is differentiable and $\nabla g$ is $1 / \beta$ -Lipschitz continuous.