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, tol: float | None = None, rtol: float | None = None, callback: Callable[[ndarray[tuple[Any, ...], dtype[_ScalarT]]], None] | None = None, show: bool = False, itershow: tuple[int, int, int] = (10, 10, 10)) → tuple[ndarray[tuple[Any, ...], dtype[_ScalarT]], ndarray[tuple[Any, ...], dtype[_ScalarT]]][source]¶

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.
tolfloat, optional: Tolerance on change of objective function (used as stopping criterion). If tol=None, run until niter is reached
rtolfloat, optional: Relative tolerance on objective function wrt initial value. Stops the solver when the ratio of the current objective function to the initial objective function is below this value. If rtol=None, run until niter is reached or the other tolerance criterion is met
callbackcallable, optional: Function with signature (callback(x)) to call after each iteration where x is the current model vector
showbool, optional: Display iterations log
itershowtuple, optional: Display set log for the first N1 steps, last N2 steps, and every N3 steps in between where N1, N2, N3 are the three element of the list.

Returns: