Power-class: The Power class.

Description

This class represents the elementwise power function \(f(x) = x^p\). If expr is a CVXR expression, then expr^p is equivalent to Power(expr, p).

Usage

Power(x, p, max_denom = 1024)
# S4 method for Power
to_numeric(object, values)
# S4 method for Power
sign_from_args(object)
# S4 method for Power
is_atom_convex(object)
# S4 method for Power
is_atom_concave(object)
# S4 method for Power
is_atom_log_log_convex(object)
# S4 method for Power
is_atom_log_log_concave(object)
# S4 method for Power
is_constant(object)
# S4 method for Power
is_incr(object, idx)
# S4 method for Power
is_decr(object, idx)
# S4 method for Power
is_quadratic(object)
# S4 method for Power
is_qpwa(object)
# S4 method for Power
.grad(object, values)
# S4 method for Power
.domain(object)
# S4 method for Power
get_data(object)
# S4 method for Power
copy(object, args = NULL, id_objects = list())
# S4 method for Power
name(x)

Arguments

x: The Expression to be raised to a power.
p: A numeric value indicating the scalar power.
max_denom: The maximum denominator considered in forming a rational approximation of p.
object: A Power object.
values: A list of numeric values for the arguments
idx: An index into the atom.
args: A list of arguments to reconstruct the atom. If args=NULL, use the current args of the atom
id_objects: Currently unused.

Methods (by generic)

to_numeric(Power): Throw an error if the power is negative and cannot be handled.
sign_from_args(Power): The sign of the atom.
is_atom_convex(Power): Is \(p \leq 0\) or \(p \geq 1\)?
is_atom_concave(Power): Is \(p \geq 0\) or \(p \leq 1\)?
is_atom_log_log_convex(Power): Is the atom log-log convex?
is_atom_log_log_concave(Power): Is the atom log-log concave?
is_constant(Power): A logical value indicating whether the atom is constant.
is_incr(Power): A logical value indicating whether the atom is weakly increasing.
is_decr(Power): A logical value indicating whether the atom is weakly decreasing.
is_quadratic(Power): A logical value indicating whether the atom is quadratic.
is_qpwa(Power): A logical value indicating whether the atom is quadratic of piecewise affine.
.grad(Power): Gives the (sub/super)gradient of the atom w.r.t. each variable
.domain(Power): Returns constraints describng the domain of the node
get_data(Power): A list containing the output of pow_low, pow_mid, or pow_high depending on the input power.
copy(Power): Returns a shallow copy of the power atom
name(Power): Returns the expression in string form.

Slots

x: The Expression to be raised to a power.

p

A numeric value indicating the scalar power.

max_denom

The maximum denominator considered in forming a rational approximation of p.

Details

For \(p = 0\), \(f(x) = 1\), constant, positive.

For \(p = 1\), \(f(x) = x\), affine, increasing, same sign as \(x\).

For \(p = 2,4,8,...\), \(f(x) = |x|^p\), convex, signed monotonicity, positive.

For \(p < 0\) and \(f(x) = \)

\(x^p\): for \(x > 0\)
\(+\infty\): \(x \leq 0\)

, this function is convex, decreasing, and positive.

For \(0 < p < 1\) and \(f(x) =\)

\(x^p\): for \(x \geq 0\)
\(-\infty\): \(x < 0\)

, this function is concave, increasing, and positive.

For \(p > 1, p \neq 2,4,8,\ldots\) and \(f(x) = \)

\(x^p\): for \(x \geq 0\)
\(+\infty\): \(x < 0\)

, this function is convex, increasing, and positive.