Convex quadratic function ($\frac{1}{2}{x}^{T}Hx+b^{T}x+c$)

A convex quadratic function on ${\mathbb{R}}^n$ has the form $q(x)=\frac{1}{2}{x}^{T}Hx+b^{T}x+c$ with a symmetric positive semidefinite matrix $H$. Its Hessian is $H$, and $q(y)-q(x)-\nabla q(x)\cdot (y-x)=\frac{1}{2}(y-x{)}^{T}H(y-x)\ge 0$, 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.