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: Optional[ndarray[tuple[Any, ...], dtype[_ScalarT]]] = None, callback: Optional[Callable[[ndarray[tuple[Any, ...], dtype[_ScalarT]]], 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