line search

Choosing how far to step along a fixed search direction by (approximately) minimizing along it.

Given a current point $x^{(k)}$ and a search direction $d^{(k)}$ , a line search chooses the step size $\alpha$ by considering the one-variable function $\varphi(\alpha)=f(x^{(k)}+\alpha d^{(k)})$ . An exact line search minimizes $\varphi$ exactly; an inexact line search (e.g. backtracking against the Armijo condition) accepts any $\alpha$ giving a "good enough" decrease, far cheaper to compute in practice.

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