pyproximal.optimization.primaldual.PrimalDual¶

pyproximal.optimization.primaldual.PrimalDual(proxf: ProxOperator, proxg: ProxOperator, A: LinearOperator, x0: ndarray[tuple[Any, ...], dtype[_ScalarT]], tau: float | ndarray[tuple[Any, ...], dtype[_ScalarT]], mu: float | ndarray[tuple[Any, ...], dtype[_ScalarT]], y0: ndarray[tuple[Any, ...], dtype[_ScalarT]] | None = None, z: ndarray[tuple[Any, ...], dtype[_ScalarT]] | None = None, theta: float = 1.0, niter: int = 10, gfirst: bool = True, callback: Callable[[...], None] | None = None, callbacky: bool = False, returny: bool = False, show: bool = False) → ndarray[tuple[Any, ...], dtype[_ScalarT]] | tuple[ndarray[tuple[Any, ...], dtype[_ScalarT]], ndarray[tuple[Any, ...], dtype[_ScalarT]]][source]¶

Primal-dual algorithm

Solves the following (possibly) nonlinear minimization problem using the general version of the first-order primal-dual algorithm of [1]:

\[\min_{\mathbf{x} \in X} g(\mathbf{Ax}) + f(\mathbf{x}) + \mathbf{z}^T \mathbf{x}\]

where \(\mathbf{A}\) is a linear operator, \(f\) and \(g\) can be any convex functions that have a known proximal operator.

This functional is effectively minimized by solving its equivalent primal-dual problem (primal in \(f\), dual in \(g\)):

\[\min_{\mathbf{x} \in X} \max_{\mathbf{y} \in Y} \mathbf{y}^T(\mathbf{Ax}) + \mathbf{z}^T \mathbf{x} + f(\mathbf{x}) - g^*(\mathbf{y})\]

where \(\mathbf{y}\) is the so-called dual variable.

Parameters:

proxfpyproximal.ProxOperator: Proximal operator of f function
proxgpyproximal.ProxOperator: Proximal operator of g function
Apylops.LinearOperator: Linear operator of g
x0numpy.ndarray: Initial vector
taufloat or numpy.ndarray: Stepsize of subgradient of \(f\). This can be constant or function of iterations (in the latter cases provided as np.ndarray)
mufloat or numpy.ndarray: Stepsize of subgradient of \(g^*\). This can be constant or function of iterations (in the latter cases provided as np.ndarray)
y0numpy.ndarray: Initial auxiliary vector. If None, set to zero
znumpy.ndarray, optional: Additional vector
thetafloat: Scalar between 0 and 1 that defines the update of the \(\bar{\mathbf{x}}\) variable - note that theta=0 is a special case that represents the semi-implicit classical Arrow-Hurwicz algorithm
niterint, optional: Number of iterations of iterative scheme
gfirstbool, optional: Apply Proximal of operator g first (True) or Proximal of operator f first (False)
callbackcallable, optional: Function with signature (callback(x)) to call after each iteration where x is the current model vector
callbackybool, optional: Modify callback signature to (callback(x, y)) when callbacky=True
returnybool, optional: Return also y
showbool, optional: Display iterations log

Returns:

xnumpy.ndarray: Inverted model
ynumpy.ndarray, optional: Inverted second model, only returned if returny=True

Notes

The Primal-dual algorithm can be expressed by the following recursion (gfirst=True):

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

where \(\tau \mu \lambda_{max}(\mathbf{A}^H\mathbf{A}) < 1\).

Alternatively for gfirst=False the scheme becomes:

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

[1]

A., Chambolle, and T., Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging”, Journal of Mathematical Imaging and Vision, 40, 8pp. 120-145. 2011.