The Pnorm class. — Pnorm-class • CVXR

This class represents the vector p-norm.

Pnorm(x, p = 2, axis = NA_real_, keepdims = FALSE, max_denom = 1024)

# S4 method for Pnorm
allow_complex(object)

# S4 method for Pnorm
to_numeric(object, values)

# S4 method for Pnorm
validate_args(object)

# S4 method for Pnorm
sign_from_args(object)

# S4 method for Pnorm
is_atom_convex(object)

# S4 method for Pnorm
is_atom_concave(object)

# S4 method for Pnorm
is_atom_log_log_convex(object)

# S4 method for Pnorm
is_atom_log_log_concave(object)

# S4 method for Pnorm
is_incr(object, idx)

# S4 method for Pnorm
is_decr(object, idx)

# S4 method for Pnorm
is_pwl(object)

# S4 method for Pnorm
get_data(object)

# S4 method for Pnorm
name(x)

# S4 method for Pnorm
.domain(object)

# S4 method for Pnorm
.grad(object, values)

# S4 method for Pnorm
.column_grad(object, value)

Arguments

x: An Expression representing a vector or matrix.
p: A number greater than or equal to 1, or equal to positive infinity.
axis: (Optional) The dimension across which to apply the function: 1 indicates rows, 2 indicates columns, and NA indicates rows and columns. The default is NA.
keepdims: (Optional) Should dimensions be maintained when applying the atom along an axis? If FALSE, result will be collapsed into an $n x 1$ column vector. The default is FALSE.
max_denom: (Optional) The maximum denominator considered in forming a rational approximation for $p$. The default is 1024.
object: A Pnorm object.
values: A list of numeric values for the arguments
idx: An index into the atom.
value: A numeric value

Details

If given a matrix variable, Pnorm will treat it as a vector and compute the p-norm of the concatenated columns.

For $p \geq 1$, the p-norm is given by $$\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$$ with domain $x \in \mathbf{R}^n$. For $p < 1, p\neq 0$, the p-norm is given by $$\|x\|_p = \left(\sum_{i=1}^n x_i^p\right)^{1/p}$$ with domain $x \in \mathbf{R}^n_+$.

Note that the "p-norm" is actually a norm only when $p \geq 1$ or $p = +\infty$. For these cases, it is convex.
The expression is undefined when $p = 0$.
Otherwise, when $p < 1$, the expression is concave, but not a true norm.

Methods (by generic)

allow_complex(Pnorm): Does the atom handle complex numbers?
to_numeric(Pnorm): The p-norm of x.
validate_args(Pnorm): Check that the arguments are valid.
sign_from_args(Pnorm): The atom is positive.
is_atom_convex(Pnorm): The atom is convex if $p \geq 1$.
is_atom_concave(Pnorm): The atom is concave if $p < 1$.
is_atom_log_log_convex(Pnorm): Is the atom log-log convex?
is_atom_log_log_concave(Pnorm): Is the atom log-log concave?
is_incr(Pnorm): The atom is weakly increasing if $p < 1$ or $p > 1$ and x is positive.
is_decr(Pnorm): The atom is weakly decreasing if $p > 1$ and x is negative.
is_pwl(Pnorm): The atom is not piecewise linear unless $p = 1$ or $p = \infty$.
get_data(Pnorm): Returns list(p, axis).
name(Pnorm): The name and arguments of the atom.
.domain(Pnorm): Returns constraints describing the domain of the node
.grad(Pnorm): Gives the (sub/super)gradient of the atom w.r.t. each variable
.column_grad(Pnorm): Gives the (sub/super)gradient of the atom w.r.t. each column variable

Slots

x: An Expression representing a vector or matrix.
p: A number greater than or equal to 1, or equal to positive infinity.
max_denom: The maximum denominator considered in forming a rational approximation for $p$.
axis: (Optional) The dimension across which to apply the function: 1 indicates rows, 2 indicates columns, and NA indicates rows and columns. The default is NA.
keepdims: (Optional) Should dimensions be maintained when applying the atom along an axis? If FALSE, result will be collapsed into an $n x 1$ column vector. The default is FALSE.
.approx_error: (Internal) The absolute difference between $p$ and its rational approximation.
.original_p: (Internal) The original input $p$.