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Local Search
Suppose we are given some optimization problem
\[ \begin{aligned} &\min & f(\mathbf x)\\ &\text{subject to}& \mathbf x &\in S \end{aligned} \]where \( S \subset \mathbb R^n \) and \( f \colon S \to \mathbb R \) . In local search, for every \( \mathbf x \in S \) , we must give some \( N_{\mathbf x} \) to be the neighborhood of \( \mathbf x \) . Then given some initial feasible solution \( \mathbf x_0 \) , for \( n = 1,2,\ldots \) , we let
\[ \mathbf x_{n} = \mathop{\rm arg\, min}_{\mathbf x \in N_{\mathbf x_{n-1}}} \left\{ f(\mathbf x)\right\} \]