α-strongly convex and β-smooth

A function is α-strongly convex and β-smooth if for all $\mathbf{x},\mathbf{y}$: $$\frac{\alpha}{2}||\mathbf{y}-\mathbf{x}||_2^2 \leq [f(\mathbf{y})-f(\mathbf{x})]-\nabla f(\mathbf{x})^\mathsf{T}(\mathbf{y}-\mathbf{x}) \leq \frac{\beta}{2}||\mathbf{y}-\mathbf{x}||_2^2$$ (multidimensional generalization)

For scalar functions, a twice-differentiable function $f$ is α-strongly convex and β-smooth if for all $x$, $$\alpha \leq f''(x) \leq \beta$$

If f is β-smooth and α-strongly convex then at any point $\mathbf{x}$, the Hessian $∇^2 f(\mathbf{x})$ satisfies: $$\alpha \mathbf{I} \preceq ∇^2 f(\mathbf{x}) \preceq \beta \mathbf{I}$$ where $\mathbf{I}$ is a $d \times d$ identity matrix.

This is the natural matrix generalization of the statement for scalar valued functions.

Note the PSD relations

Equivalently for any $\mathbf{z}$,

#incomplete

condition number

$\kappa = \frac{\beta}{\alpha}$ is called the condition number of $f$