picos.expressions.exp_powtrace¶

Implements PowerTrace.

Outline¶

Classes¶

The trace of the $p$ -th power of a hermitian matrix.

Classes¶

PowerTrace¶

class picos.expressions.exp_powtrace.PowerTrace(x, p, m=None, denominator_limit=1000)[source]¶

Bases: picos.expressions.expression.Expression

The trace of the $p$ -th power of a hermitian matrix.

Definition

Let $p \in \mathbb{Q}$ .

If the base expressions is a real scalar $x$ and no additional constant $m$ is given, then this is the power $x^p$ .
If the base expressions is a real scalar $x$ , $p \in [0, 1]$ , and a positive scalar constant $m$ is given, then this is the scaled power $m x^p$ .
If the base expression is a hermitian matrix $X$ and no additional constant $M$ is given, then this is the trace of power $\operatorname{tr}(X^p)$ .
If the base expression is a hermitian matrix $X$ , $p \in [0, 1]$ , and a hermitian positive semidefinite constant matrix $M$ of same shape as $X$ is given, then this is the trace of a scaled power $\operatorname{tr}(M X^p)$ .

No other case is supported. In particular, if $p \not\in [0, 1]$ , then $m$ / $M$ must be undefined (None).

Warning

For a constraint of the form $x^p \leq t$ with $p < 1$ and $p \neq 0$ , PICOS enforces $x \geq 0$ during solution search.
For a constraint of the form $\operatorname{tr}(X^p) \leq t$ or $\operatorname{tr}(M X^p) \leq t$ with $p < 1$ and $p \neq 0$ , PICOS enforces $X \succeq 0$ during solution search.
For a constraint of the form $\operatorname{tr}(X^p) \leq t$ or $\operatorname{tr}(M X^p) \leq t$ with $p > 1$ , PICOS enforces $t \geq 0$ during solution search.

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

Construct a PowerTrace.

Parameters

x (AffineExpression) – The scalar or symmetric matrix to form a power of.
p (float) – The value for $p$ , which is cast to a limited precision fraction.
m (AffineExpression or anything recognized by load_data) – An additional positive semidefinite constant to multiply the power with.
denominator_limit (int) – The largest allowed denominator when casting $p$ to a fraction. Higher values can yield a greater precision at reduced performance.

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

property m¶: An additional factor to multiply the power with.

property n¶: Diagonal length of x.

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

property p¶

The parameter $p$ .

This is a limited precision version of the paramter used when the expression was constructed.

property x¶: The matrix concerned.