picos.expressions.exp_norm¶

Implements Norm.

Classes

class picos.expressions.exp_norm.Norm(x, p=2, q=None, denominator_limit=1000)[source]¶

Bases: Expression

Entrywise $p$ -norm or $L_{p,q}$ -norm of an expression.

This class can represent the absolute value, the modulus, a vector $p$ -norm and an entrywise matrix $L_{p,q}$ -norm of a (complex) affine expression. In addition to these convex norms, it can represent the concave generalized vector $p$ -norm with $0 < p \leq 1$ .

Not all of these norms are available on the complex field; see the definitions below to learn more.

Definition

If $q$ is not given (None), then it is set equal to $p$ .

If the normed expression is a real scalar $x$ , then this is the absolute value

$|x| = \begin{cases} x, &\text{if }x \geq 0,\\ -x, &\text{otherwise.} \end{cases}$

The parameters $p$ and $q$ are ignored in this case.
If the normed expression is a complex scalar $z$ , then this is the modulus

$|z| = \sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}.$

The parameters $p$ and $q$ are ignored in this case.
If the normed expression is a real vector $x$ and $p = q$ , then this is the (generalized) vector $p$ -norm

$\lVert x \rVert_p = \left(\sum_{i=1}^n |x_i|^p\right)^{\frac{1}{p}}$

for $p \in \mathbb{Q} \cup \{\infty\}$ with $p > 0$ and $|x_i|$ the absolute value of the $i$ -th entry of $x$ .

Note that for $p < 1$ the expression is not convex and thus not a proper norm. However, it is concave over the nonnegative orthant and posing a lower bound on such a generalized norm yields a convex constraint for $x \geq 0$ .

Warning

When you pose a lower bound on a concave generalized norm ( $p < 1$ ), then PICOS enforces $x \geq 0$ through an auxiliary constraint during solution search.

Special cases:
- For $p = 1$ , this is the Manhattan or Taxicab norm $\lVert x \rVert_{\text{sum}}$ .
- For $p = 2$ , this is the Euclidean norm $\lVert x \rVert = \lVert x \rVert_2$ .
- For $p = \infty$ , this is the Maximum, Chebyshev, or Infinity norm $\lVert x \rVert_{\text{max}}$ .
If the normed expression is a real vector $x$ and $p \neq q$ , then it is treated as a matrix with a single row or a single column, depending on the shape associated with $x$ . See case (5).
If the normed expression is a complex vector $z$ and $p = q$ , then the definition is the same as in case (3) but with $x = z$ , $|x_i|$ the modulus of the $i$ -th entry of $x$ , and $p \geq 1$ .
If the normed expression is a real $m \times n$ matrix $X$ , then this is the $L_{p,q}$ -norm

$\lVert X \rVert_{p,q} = \left(\sum_{j = 1}^n \left(\sum_{i = 1}^m |X_{ij}|^p \right)^{\frac{q}{p}} \right)^{\frac{1}{q}}$

for $p, q \in \mathbb{Q} \cup \{\infty\}$ with $p,q \geq 1$ .

If $p = q$ , then this is equal to the (generalized) vector $p$ -norm of the the vectorized matrix, that is $\lVert X \rVert_{p,p} = \lVert \operatorname{vec}(X) \rVert_p$ . In this case, the requirement $p \geq 1$ is relaxed to $p > 0$ and $X$ may be a complex matrix. See case (3).

Special cases:
- For $p = q = 2$ , this is the Frobenius norm $\lVert X \rVert = \lVert X \rVert_F$ .
- For $p = 1$ , $q = \infty$ , this is the maximum absolute column sum
  
  $\lVert X \rVert_{1,\infty} = \max_{j=1 \ldots n} \sum_{i=1}^m |X_{ij}|.$
  
  This equals the operator norm induced by the vector $1$ -norm. You can obtain the maximum absolute row sum (the operator norm induced by the vector $\infty$ -norm) by first transposing $X$ .
Complex matrix norms are not supported.

Note

You can write $\infty$ in Python as float("inf").

__ge__(other)[source]¶: Return a constraint that the expression is lower-bounded.

__init__(x, p=2, q=None, denominator_limit=1000)[source]¶

Construct a Norm.

Parameters

x (ComplexAffineExpression) – The affine expression to take the norm of.
p (float) – The value for $p$ , which is cast to a limited precision fraction.
q (float) – The value for $q$ , which is cast to a limited precision fraction. The default of None means equal to $p$ .
denominator_limit (int) – The largest allowed denominator when casting $p$ and $q$ to a fraction. Higher values can yield a greater precision at reduced performance.

__le__(other)[source]¶: Return a constraint that the expression is upper-bounded.

__mul__(other)[source]¶: Denote multiplication with another expression on the right.

__pow__(other)[source]¶: Denote exponentiation with another, scalar expression.

__rmul__(other)[source]¶: Denote multiplication with another expression on the left.

property p¶

The parameter $p$ .

This is a limited precision version of the parameter used when the norm was constructed.

property pden¶: The limited precision fraction denominator of $p$ .

property pnum¶: The limited precision fraction numerator of $p$ .

property q¶

The parameter $q$ .

This is a limited precision version of the parameter used when the norm was constructed.

property qden¶: The limited precision fraction denominator of $q$ .

property qnum¶: The limited precision fraction numerator of $q$ .

property x¶: Real expression whose norm equals that of the original expression.

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