Class NesterovFunction

Parameters
----------
l : Non-negative float. The Lagrange multiplier, or regularisation
        constant, of the function.

A : A (usually sparse) array. The linear operator for the Nesterov
        formulation. May not be None!

mu: Non-negative float. The regularisation constant for the smoothing.

penalty_start : Non-negative integer. The number of columns, variables
        etc., to except from penalisation. Equivalently, the first
        index to be penalised. Default is 0, all columns are included.

Overrides: object.__init__

Returns the smoothed function value.

Parameters
----------
beta : Numpy array. A weight vector.

mu : Non-negative float. The regularisation constant for the smoothing.

Function value with known alpha.

Decorators:

• @abc.abstractmethod

Gradient of the function at beta.

Parameters
----------
beta : Numpy array. The point at which to evaluate the gradient.

Overrides: Gradient.grad

Set the regularisation constant for the smoothing.

Parameters
----------
mu : Non-negative float. The regularisation constant for the smoothing
        to use from now on.

Returns
-------
old_mu : Non-negative float. The old regularisation constant for the
        smoothing that was overwritten and no longer is used.

Dual variable of the Nesterov function.

Parameters
----------
beta : Numpy array (p-by-1). The variable for which to compute the dual
        variable alpha.

Linear operator of the Nesterov function multiplied by the corresponding Lagrange multipliers.

Specialise this function if you need to. E.g. if you are smoothing a sum of functions with different Lagrange multipliers.

Compute A'*alpha.

Parameters
----------
alpha : List of numpy arrays (x-by-1). The dual variable alpha.

Projection onto the compact space of the Nesterov function.

Parameters
----------
alpha : List of numpy arrays (x-by-1). The not-yet-projected dual
        variable alpha.

Decorators:

• @abc.abstractmethod

The maximum value of the regularisation of the dual variable. We
have

    M = max_{alpha in K} 0.5*|alpha|²_2.

Decorators:

• @abc.abstractmethod

Compute a "good" value of mu with respect to the given beta.

Parameters
----------
beta : Numpy array (p-by-1). The primal variable at which to compute a
        feasible value of mu.

Largest eigenvalue of the corresponding covariance matrix.

From the interface "Eigenvalues".

Overrides: Eigenvalues.lambda_max

Lipschitz constant of the gradient.

From the interface "LipschitzContinuousGradient".

Overrides: LipschitzContinuousGradient.L

The proximal operator corresponding to this function.

The proximal operator is computed numerically. This method should be
overloaded if the function has a known proximal operator.

From the interface "ProximalOperator".

Parameters
----------
beta : Numpy array (p-by-1). The point at which to apply the proximal
        operator.

factor : Positive float. A factor by which the Lagrange multiplier is
        scaled. This is usually the step size.

eps : Positive float. This is the stopping criterion for inexact
        proximal methods, where the proximal operator is approximated
        numerically.

max_iter : Positive integer. This is the maximum number of iterations
        for inexact proximal methods, where the proximal operator is
        approximated numerically.

Overrides: ProximalOperator.prox