qics.cones.OpPerspecTr

class qics.cones.OpPerspecTr(n, func, iscomplex=False)

A class representing a trace operator perspective epigraph cone

$${\mathcal{OPT}}_{n,g}=\text{cl}\{(t,X,Y)\in \mathbb{R}\times {\mathbb{H}}_{++}^n\times {\mathbb{H}}_{++}^n:t\ge \text{tr}[P_g(X,Y)]\},$$

for an operator concave function $g:(0,\mathrm{\infty })\to \mathbb{R}$, where

$$P_g(X,Y)=X^{1/2}g(X^{-1/2}Y{X}^{-1/2})X^{1/2},$$

is the operator perspective of $g$.

Parameters:

nint

Dimension of the matrices $X$ and $Y$.

funcstring or float

Choice for the function $g$. Can be defined in the following ways.

• $g(x)=-\log (x)$ if func="log"

• $g(x)=-x^p$ if func=p is a float where $p\in (0,1)$

• $g(x)=x^p$ if func=p is a float where $p\in [-1,0)\cup (1,2)$

iscomplexbool

Whether the matrices $X$ and $Y$ are defined over ${\mathbb{H}}^n$ (True), or restricted to ${\mathbb{S}}^n$ (False). The default is False.

See also

OpPerspecEpi

Operator perspective epigraph

Notes

We do not support operator perspectives for p=0, p=1, and p=2 as these functions are more efficiently modelled using just the positive semidefinite cone.

• When $g(x)=x^0$, $P_g(X,Y)=X$.

• When $g(x)=x^1$, $P_g(X,Y)=Y$.

• When $g(x)=x^2$, $P_g(X,Y)=Y{X}^{-1}Y$, which can be modelled using the Schur complement lemma, i.e., if $X\succ 0$, then

  $$\begin{array}{r}\left[\begin{array}{cc}X& Y\\ Y& T\end{array}\right]⪰0⟺T⪰Y{X}^{-1}Y.\end{array}$$