Convex quadratic function
= Convex quadratic function
{title2=$\tfrac12x^THx+b^Tx+c$}
A <convex quadratic function> on $\mathbb R^n$ has the form $q(x)=\tfrac12x^THx+b^Tx+c$ with a symmetric <positive semidefinite matrix> $H$. Its Hessian is $H$, and $q(y)-q(x)-\nabla q(x)\cdot(y-x)=\tfrac12(y-x)^TH(y-x)\geq0$, proving convexity directly. A stationary point is a global minimum. On a compact polytope, a minimum either is stationary in the interior or minimizes a restriction to a boundary face.