pyproximal.optimization.primal.LinearizedADMM

pyproximal.optimization.primal.LinearizedADMM(proxf: ProxOperator, proxg: ProxOperator, A: LinearOperator, x0: ndarray[tuple[Any, ...], dtype[_ScalarT]], tau: float, mu: float, niter: int = 10, z0: ndarray[tuple[Any, ...], dtype[_ScalarT]] | None = None, callback: Callable[[ndarray[tuple[Any, ...], dtype[_ScalarT]]], None] | None = None, show: bool = False) → tuple[ndarray[tuple[Any, ...], dtype[_ScalarT]], ndarray[tuple[Any, ...], dtype[_ScalarT]]]

Linearized Alternating Direction Method of Multipliers

Solves the following minimization problem using Linearized Alternating Direction Method of Multipliers:

\[\mathbf{x} = \argmin_\mathbf{x} f(\mathbf{x}) + g(\mathbf{A}\mathbf{x})\]

where \(f(\mathbf{x})\) and \(g(\mathbf{x})\) are any convex function that has a known proximal operator and \(\mathbf{A}\) is a linear operator.

Parameters:

proxfpyproximal.ProxOperator

Proximal operator of f function

proxgpyproximal.ProxOperator

Proximal operator of g function

Apylops.LinearOperator

Linear operator

x0numpy.ndarray

Initial vector

taufloat, optional

Positive scalar weight, which should satisfy the following condition to guarantee convergence: \(\mu \in (0, \tau/\lambda_{max}(\mathbf{A}^H\mathbf{A})]\).

mufloat, optional

Second positive scalar weight, which should satisfy the following condition to guarantees convergence: \(\mu \in (0, \tau/\lambda_{max}(\mathbf{A}^H\mathbf{A})]\).

niterint, optional

Number of iterations of iterative scheme

z0numpy.ndarray

Initial auxiliary vector. If None, initialized to A @ x0.

callbackcallable, optional

Function with signature (callback(x)) to call after each iteration where x is the current model vector

showbool, optional

Display iterations log

Returns:

xnumpy.ndarray

Inverted model

znumpy.ndarray

Inverted second model

Raises:

ValueError

If both x0 and z0 are set to None

See also

ADMM

ADMM

ADMML2

ADMM with L2 misfit function

Notes

The Linearized-ADMM algorithm can be expressed by the following recursion [1]:

\[\begin{split}\mathbf{x}^{k+1} = \prox_{\mu f}(\mathbf{x}^{k} - \frac{\mu}{\tau} \mathbf{A}^H(\mathbf{A} \mathbf{x}^k - \mathbf{z}^k + \mathbf{u}^k))\\ \mathbf{z}^{k+1} = \prox_{\tau g}(\mathbf{A} \mathbf{x}^{k+1} + \mathbf{u}^k)\\ \mathbf{u}^{k+1} = \mathbf{u}^{k} + \mathbf{A}\mathbf{x}^{k+1} - \mathbf{z}^{k+1}\end{split}\]

[1]

N., Parikh, “Proximal Algorithms”, Foundations and Trends in Optimization. 2013.

Examples using pyproximal.optimization.primal.LinearizedADMM

Denoising

Denoising

Relaxed Mumford-Shah regularization

Relaxed Mumford-Shah regularization