Square-to-Linear Function and Properties

The square-to-linear function in $mathbf{R}^n$ is the function $f : mathbf{R}^n times mathbf{R} rightarrow mathbf{R}$ , with domain the set

$mbox{bf dom} f = left{ (x,y) in mathbf{R}^n times mathbf{R} ~:~ y > 0 right}$

and values given by

$f(x) = left{ begin{array}{ll} displaystylefrac{x^Tx}{y} & mbox{if } y>0, +infty & mbox{otherwise.} end{array}right.$

This function is convex, since its domain is, and inside the interior of the domain, the Hessian is given by

$nabla^2 f(x) = frac{2}{y^3} left( begin{array}{cc} y^2 I & -yx -yx^T & x^Tx end{array} right) = .$

We check that the Hessian is positive semi-definite: for any $w = (z,t) in mathbf{R}^{n} times mathbf{R}$ , we have

$frac{y^3}{2} w^T nabla^2 f(x) w = left( begin{array}{c} z t end{array} right)^T left( begin{array}{cc} y^2 I & -yx -yx^T & x^Tx end{array} right) left( begin{array}{c} z t end{array} right) = |yz - t x|_2^2 ge 0.$