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@jemoka / Jemoka Knowledge Base / wiki/concepts/iterative_method.md
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--- title: "iterative method" type: concept related: [Descent Method] source: https://www.jemoka.com/posts/kbhiterative_method/ confidence: high status: active --- 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} various iterative methods descent method