QuadraticForm.convex_conj

property QuadraticForm.convex_conj: The convex conjugate functional of the quadratic form.

Notes

The convex conjugate of the quadratic form $<x, Ax> + <b, x> + c$ is given by

$(<x, Ax> + <b, x> + c)^* (x) = <(x - b), A^-1 (x - b)> - c = <x , A^-1 x> - <x, A^-* b> - <x, A^-1 b> + <b, A^-1 b> - c.$

If the quadratic part of the functional is zero it is instead given by a translated indicator function on zero, i.e., if

$f(x) = <b, x> + c,$

then

$f^*(x^*) = \begin{cases} -c & \text{if } x^* = b \\ \infty & \text{else.} \end{cases}$

See Also

IndicatorZero

Notes