Square-to-Linear Function and PropertiesThe 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 \[ f(x) = \left\{ \begin{array}{ll} \displaystyle\frac{x^Tx}{y} & \mbox{if } y>0, \\+\infty & \mbox{otherwise.} \end{array}\right. \] 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 \[ \frac{y^3}{2} w^T \nabla^2 f(x) w = \left( \begin{array}{c} z \\ t \end{array} \right)^T \left( \begin{array}{cc} y^2 I & -yx \\ -yx^T & x^Tx \end{array} \right) \left( \begin{array}{c} z \\ t \end{array} \right) = \|yz - t x\|_2^2 \ge 0. \] |