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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

## Algorithm

## See also

## References

Usually it is undesirable to exactly minimize the function $displaystyle\; phi(alpha)$ 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.)

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

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

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

- $j=j+1.,$

- iii) Return $scriptstyle\; alpha=alpha^j.,$

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

- J. Nocedal and S. J. Wright, Numerical optimization. Springer Verlag, New York, NY, 1999.

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Last updated on Wednesday August 20, 2008 at 17:25:18 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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