Basic continuity and geometric properties:
Convex: \(f(x)\) is convex iff \(\forall u, v \in \Omega. f(u) \geq f(v) + \langle u-v, \nabla f(v) \rangle\). If the inequality is strict, then it’s strictly convex.
Strongly convex: \(f(x)\) is \(m\)-strongly convex iff \(\forall u, v \in \Omega. f(u) \geq f(v) + \langle u-v, \nabla f(v)\rangle + \frac{m}{2}\lVert u-v \rVert^2\). It bends more strongly than some quadratic times an \(m\).
Lipschitz: \(f(x)\) is \(L\)-Lipschitz iff \(\lVert \nabla f(x) \rVert \leq L \iff \lVert f(u) - f(v) \rVert \leq L \lVert u - v \rVert\).
Smoothness: \(f(x)\) is \(\beta\)-smooth if its gradient is \(\beta\)-Lipschitz, i.e., \(\lVert \nabla f(u) - \nabla f(v) \rVert \leq \beta\lVert u-v\rVert \iff f(u) \leq f(u) + \frac{\beta}{2} \lVert u-v \rVert^2\).
If \(f(x)\) is \(L\)-Lipschitz,