p-norm

class cvxpy.pnorm(x, p=2, axis=None, max_denom=1024)

The vector p-norm.

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

$$\|x\|_p = \left(\sum_i |x_i|^p \right)^{1/p},$$

with domain $x \in \mathbf{R}^n$.

For $p < 1,\ p \neq 0$, the p-norm is given by

$$\|x\|_p = \left(\sum_i x_i^p \right)^{1/p},$$

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 not defined when $p = 0$.
• Otherwise, when $p < 1$, the expression is concave, but it is not a true norm.

Note

Generally, p cannot be represented exactly, so a rational, i.e., fractional, approximation must be made.

Internally, pnorm computes a rational approximation to the reciprocal $1/p$ with a denominator up to max_denom. The resulting approximation can be found through the attribute pnorm.p. The approximation error is given by the attribute pnorm.approx_error. Increasing max_denom can give better approximations.

When p is an int or Fraction object, the approximation is usually exact.

Parameters:

x : cvxpy.Variable

The value to take the norm of.

p : int, float, Fraction, or string

If p is an int, float, or Fraction then we must have $p \geq 1$.

The only other valid inputs are numpy.inf, float('inf'), float('Inf'), or the strings "inf" or "inf", all of which are equivalent and give the infinity norm.

max_denom : int

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

axis : 0 or 1

The axis to apply the norm to.

Returns:

Expression :

An Expression representing the norm.

static graph_implementation(arg_objs, shape, data=None)

Reduces the atom to an affine expression and list of constraints.

Parameters:

arg_objs : list

LinExpr for each argument.

shape : tuple

The shape of the resulting expression.

data : :

Additional data required by the atom.

Returns:

tuple :

(LinOp for objective, list of constraints)

Notes

Implementation notes.

• For general $p \geq 1$, the inequality $\|x\|_p \leq t$ is equivalent to the following convex inequalities:

  $$\begin{split}|x_i| &\leq r_i^{1/p} t^{1 - 1/p}\\ \sum_i r_i &= t.\end{split}$$

  These inequalities happen to also be correct for $p = +\infty$, if we interpret $1/\infty$ as $0$.

• For general $0 < p < 1$, the inequality $\|x\|_p \geq t$ is equivalent to the following convex inequalities:

  $$\begin{split}r_i &\leq x_i^{p} t^{1 - p}\\ \sum_i r_i &= t.\end{split}$$

• For general $p < 0$, the inequality $\|x\|_p \geq t$ is equivalent to the following convex inequalities:

  $$\begin{split}t &\leq x_i^{-p/(1-p)} r_i^{1/(1 - p)}\\ \sum_i r_i &= t.\end{split}$$

Although the inequalities above are correct, for a few special cases, we can represent the p-norm more efficiently and with fewer variables and inequalities.

• For $p = 1$, we use the representation

  $$\begin{split}x_i &\leq r_i\\ -x_i &\leq r_i\\ \sum_i r_i &= t\end{split}$$

• For $p = \infty$, we use the representation

  $$\begin{split}x_i &\leq t\\ -x_i &\leq t\end{split}$$

  Note that we don’t need the $r$ variable or the sum inequality.

• For $p = 2$, we use the natural second-order cone representation

  $$\|x\|_2 \leq t$$

  Note that we could have used the set of inequalities given above if we wanted an alternate decomposition of a large second-order cone into into several smaller inequalities.