1st order condition

differentiable \(f\) with convex domain IFF:

\begin{equation} f\qty(y) \geq f\qty(x) + \nabla f\qty(x)^{T} \qty(y-x), \forall x,y \in \text{dom } f \end{equation}

“the function is everywhere above the Taylor approximation” => “first-order Taylor approximation of \(f\) is a global underestimator of \(f\).”

2nd order condition

for twice differentiable \(f\) with convex domain, we have:

• \(f\) is convex IFF \(\nabla^{2} f\qty(x) \succeq 0, \forall x \in \text{dom } f\) (i.e. that the Hessian is PSD)
• if \(\nabla^{2} f\qty(x) \succ 0 \forall x \in \text{dom } f\), then we call \(f\) strictly convex (i.e. that the Hessian is PD)

you may enjoy using Cauchy-Schwartz Inequality to show these.