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.