strongly convex function

f is strongly convex with parameter m>0 if f(x)-\tfrac{m}{2}\|x\|^2 is convex, equivalently (for twice differentiable f) if every eigenvalue of the Hessian H(x) is \ge m at every point. Strong convexity is strictly stronger than strict convexity -- f(x)=x^4 is strictly convex everywhere but not strongly convex, since f'(0)=0 admits no positive floor -- and it is what guarantees a specific convergence rate for gradient-based methods, not merely eventual convergence.