adelie.adelie_core.constraint.ConstraintBase64#

class adelie.adelie_core.constraint.ConstraintBase64#

Base constraint class.

The purpose of a constraint class is to define methods that update certain quantities that are required for solving the constrained group lasso problem.

Every constraint-like class must inherit from this class and override the methods before passing into the solver.

Methods

`__init__`(self)
`buffer_size`(self)	Returns the buffer size in unit of 8 bytes.
`clear`(self)	Clears internal data.
`dual`(self, arg0, arg1)	Returns the current dual variable in sparse format.
`duals`(self)	Returns the number of dual variables.
`duals_nnz`(self)	Returns the number of non-zero dual values.
`gradient`(self, x, out)	Computes the gradient of the Lagrangian.
`gradient_static`(self, x, mu, out)	Computes the gradient of the Lagrangian.
`primals`(self)	Returns the number of primal variables.
`project`(self, x)	Computes a projection onto the feasible set.
`solve`(self, x, quad, linear, l1, l2, Q, buffer)	Computes the block-coordinate update.
`solve_zero`(self, arg0, arg1)	Solves the zero primal KKT condition problem.

Attributes

`dual_size`	Number of duals.
`primal_size`	Number of primals.

__init__(self: adelie.adelie_core.constraint.ConstraintBase64) → None#

buffer_size(self: adelie.adelie_core.constraint.ConstraintBase64) → int#

Returns the buffer size in unit of 8 bytes.

Returns:

sizeint: Buffer size in unit of 8 bytes.

clear(self: adelie.adelie_core.constraint.ConstraintBase64) → None#

Clears internal data.

The state of the constraint object must return back to that of the initial construction.

dual(self: adelie.adelie_core.constraint.ConstraintBase64, arg0: numpy.ndarray[numpy.int64[1, n], flags.writeable], arg1: numpy.ndarray[numpy.float64[1, n], flags.writeable]) → None#

Returns the current dual variable in sparse format.

Parameters:

indices(nnz,) ndarray: The indices with non-zero dual values. The size must be at least the value returned by duals_nnz().
values(nnz,) ndarray: The non-zero dual values corresponding to indices. The size must be at least the value returned by duals_nnz().

duals(self: adelie.adelie_core.constraint.ConstraintBase64) → int#

Returns the number of dual variables.

Returns:

sizeint: Number of dual variables.

duals_nnz(self: adelie.adelie_core.constraint.ConstraintBase64) → int#

Returns the number of non-zero dual values.

Returns:

nnzint: Number of non-zero dual values.

gradient(self: adelie.adelie_core.constraint.ConstraintBase64, x: numpy.ndarray[numpy.float64[1, n]], out: numpy.ndarray[numpy.float64[1, n], flags.writeable]) → None#

Computes the gradient of the Lagrangian.

The gradient of the Lagrangian (with respect to the primal) is given by

\[\begin{align*} \mu^\top \phi'(x) \end{align*}\]

where \(\phi'(x)\) is the Jacobian of \(\phi\) at \(x\) and \(\mu\) is the dual solution from the last call to solve().

Parameters:

x(d,) ndarray: The primal \(x\) at which to evaluate the gradient.
out(d,) ndarray: The output vector to store the gradient.

gradient_static(self: adelie.adelie_core.constraint.ConstraintBase64, x: numpy.ndarray[numpy.float64[1, n]], mu: numpy.ndarray[numpy.float64[1, n]], out: numpy.ndarray[numpy.float64[1, n], flags.writeable]) → None#

Computes the gradient of the Lagrangian.

The gradient of the Lagrangian (with respect to the primal) is given by

\[\begin{align*} \mu^\top \phi'(x) \end{align*}\]

where \(\phi'(x)\) is the Jacobian of \(\phi\) at \(x\) and \(\mu\) is the dual given by mu.

Parameters:

x(d,) ndarray: The primal \(x\) at which to evaluate the gradient.
mu(m,) ndarray: The dual \(\mu\) at which to evaluate the gradient.
out(d,) ndarray: The output vector to store the gradient.

primals(self: adelie.adelie_core.constraint.ConstraintBase64) → int#

Returns the number of primal variables.

Returns:

sizeint: Number of primal variables.

project(self: adelie.adelie_core.constraint.ConstraintBase64, x: numpy.ndarray[numpy.float64[1, n], flags.writeable]) → None#

Computes a projection onto the feasible set.

The feasible set is defined by \(\{x : \phi(x) \leq 0 \}\). A projection can be user-defined, that is, the user may define any norm \(\|\cdot\|\) such that the function returns a solution to

\[\begin{split}\begin{align*} \mathrm{minimize}_z \quad& \|x - z\| \\ \text{subject to} \quad& \phi(z) \leq 0 \end{align*}\end{split}\]

This function is only used by the solver after convergence to attempt to bring the coordinates into the feasible set. If not overriden, it will perform a no-op, assuming \(x\) is already feasible.

Parameters:

x(d,) ndarray: The primal \(x\) to project onto the feasible set. The output is stored back in this argument.

solve(self: adelie.adelie_core.constraint.ConstraintBase64, x: numpy.ndarray[numpy.float64[1, n], flags.writeable], quad: numpy.ndarray[numpy.float64[1, n]], linear: numpy.ndarray[numpy.float64[1, n]], l1: float, l2: float, Q: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], buffer: numpy.ndarray[numpy.uint64[1, n], flags.writeable]) → None#

Computes the block-coordinate update.

The block-coordinate update is given by solving

\[\begin{split}\begin{align*} \mathrm{minimize}_x \quad& \frac{1}{2} x^\top \Sigma x - v^\top x + \lambda_1 \|x\|_2 + \frac{\lambda_2}{2} \|x\|_2^2 \\ \text{subject to} \quad& \phi(Q x) \leq 0 \end{align*}\end{split}\]

where \(\phi\) defines the current constraint.

Parameters:

x(d,) ndarray: The primal \(x\). The passed-in values may be used as a warm-start for the internal solver. The output is stored back in this argument.
quad(d,) ndarray: The quadratic component \(\Sigma\).
linear(d,) ndarray: The linear component \(v\).
l1float: The first regularization \(\lambda_1\).
l2float: The second regularization \(\lambda_2\).
Q(d, d) ndarray: Orthogonal matrix \(Q\).
buffer(b,) ndarray: Buffer of type uint64_t aligned at 8 bytes. The size must be at least as large as buffer_size().

solve_zero(self: adelie.adelie_core.constraint.ConstraintBase64, arg0: numpy.ndarray[numpy.float64[1, n]], arg1: numpy.ndarray[numpy.uint64[1, n], flags.writeable]) → float#

Solves the zero primal KKT condition problem.

The zero primal KKT condition problem is given by

\[\begin{align*} \mathrm{minimize}_{\mu \geq 0} \|v - \phi'(0)^\top \mu\|_2 \end{align*}\]

where \(\phi\) is the current constraint function and \(\mu\) is the dual variable. It is advised, but not necessary, that the object stores the solution internally so that a subsequent call to dual() will return the solution.

Parameters:

v(d,) ndarray: The vector \(v\).
buffer(b,) ndarray: Buffer of type uint64_t aligned at 8 bytes. The size must be at least as large as buffer_size().

Returns:

normfloat: The optimal objective for the zero primal KKT condition problem.

dual_size#: Number of duals.

primal_size#: Number of primals.