pyproximal.optimization.primal.HQS¶

pyproximal.optimization.primal.HQS(proxf: ProxOperator, proxg: ProxOperator, x0: ndarray[tuple[Any, ...], dtype[_ScalarT]], tau: float | ndarray[tuple[Any, ...], dtype[_ScalarT]], niter: int = 10, z0: ndarray[tuple[Any, ...], dtype[_ScalarT]] | None = None, gfirst: bool = True, tol: float | None = None, rtol: float | None = None, callback: Callable[[...], None] | None = None, callbackz: bool = False, show: bool = False, itershow: tuple[int, int, int] = (10, 10, 10)) → tuple[ndarray[tuple[Any, ...], dtype[_ScalarT]], ndarray[tuple[Any, ...], dtype[_ScalarT]]][source]¶

Half Quadratic splitting

Solves the following minimization problem using Half Quadratic splitting algorithm:

\[\begin{split}\mathbf{x},\mathbf{z} = \argmin_{\mathbf{x},\mathbf{z}} f(\mathbf{x}) + g(\mathbf{z}) \\ s.t. \; \mathbf{x}=\mathbf{z}\end{split}\]

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

Parameters:

proxfpyproximal.ProxOperator: Proximal operator of f function
proxgpyproximal.ProxOperator: Proximal operator of g function
x0numpy.ndarray: Initial vector (not required when gfirst=False, can pass None)
taufloat or numpy.ndarray: Positive scalar weight, which should satisfy the following condition to guarantees convergence: \(\tau \in (0, 1/L]\) where L is the Lipschitz constant of \(\nabla f\). Finally note that \(\tau\) can be chosen to be a vector of size niter such that different \(\tau\) is used at different iterations (i.e., continuation strategy)
niterint: Number of iterations of iterative scheme
z0numpy.ndarray, optional: Initial z vector (not required when gfirst=True)
gfirstbool, optional: Apply Proximal of operator g first (True) or Proximal of operator f first (False)
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
callbackzbool, optional: Modify callback signature to (callback(x, z)) when callbackz=True
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: