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, tol: float | None = None, rtol: float | None = None, callback: Callable[[...], None] | None = None, callbacky: bool = False, returny: bool = False, show: bool = False, itershow: tuple[int, int, int] = (10, 10, 10)) → 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:

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

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 numpy.ndarray)
mufloat or numpy.ndarray: Stepsize of subgradient of \(g^*\). This can be constant or function of iterations (in the latter cases provided as numpy.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)
tolfloat, optional: Tolerance on x/y updates (used as stopping criterion). If tol=None, run until niter is reached or the other tolerance criterion is met
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
callbackybool, optional: Modify callback signature to (callback(x, y)) when callbacky=True
returnybool, optional: Return also y
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: