constituents
requirements
Iterative methods require a starting point \(x^{(0)}\) such that:
- \(x^{(0)} \in \text{dom } f\)
- sublevel set \(S = \qty {x \mid f\qty(x) \leq f\qty(x^{(0)})}\) is closed
additional information
strong convexity
\(f\) is strongly convex on \(S\) if there exists \(m > 0\) such that:
\begin{align} \nabla^{2}f\qty(x) \succeq mI, \forall x \in S \end{align}
if \(f\) is strongly convex for \(x, y \in S\), we have:
\begin{align} f\qty(y) \geq f\qty(x) + \nabla f\qty(x)^{T} \qty(y-x) + \frac{m}{2} \norm{x-y}_{2}^{2} \end{align}
and thus we can get a nice convergence criteria:
\begin{align} f\qty(x) - p^{*} \leq \frac{1}{2m}\norm{\nabla f\qty(x)}_{2}^{2} \end{align}