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--- title: "function convexity conditions" source: https://www.jemoka.com/posts/kbhfunction_convexity_conditions/ --- 1st order condition differentiable \(f\) with convex domain is convex 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. We also call the PSD-ness of \(\nabla^{2}f\qty(x)\) the “curvature.”