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Backtracking line search

In (unconstrained) optimization, the backtracking linesearch strategy is used as part of a linesearch method, to compute how far one should move along a given search direction.

Motivation

Usually it is undesirable to exactly minimize the function $displaystyle phi\left(alpha\right)$ in the generic linesearch algorithm. One way to inexactly minimize $displaystyle phi$ is by finding an $displaystyle alpha_k$ that gives a sufficient decrease in the objective function $f:mathbb R^ntomathbb R$ (assumed smooth), in the sense of the Armijo condition holding. This condition, when used appropriately as part of a backtracking linesearch, is enough to generate an acceptable step length. (It is not sufficient on its own to ensure that a reasonable value is generated, since all $displaystyle alpha$ small enough will satisfy the Armijo condition. To avoid the selection of steps that are too short, the additional curvature condition is usually imposed.)

Algorithm

i) Set iteration counter $scriptstyle j,=,0$. Make an initial guess $scriptstyle alpha^j,>,0$ and choose some $scriptstyle tau,in,\left(0,1\right).,$

ii) Until $scriptstyle alpha^j,$ satisfies the Armijo condition:

$alpha^\left\{j+1\right\}=taualpha^j,,$

$j=j+1.,$

iii) Return $scriptstyle alpha=alpha^j.,$

In other words, reduce $scriptstyle alpha^0$ geometrically, with rate $scriptstyletau,$, until the Armijo condition holds.