Severi–Brauer variety

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

Motivation

Usually it is undesirable to exactly minimize the function $\displaystyle \phi (\alpha )$ 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 $f:\mathbb {R} ^{n}\to \mathbb {R}$ (assumed smooth), in the sense of the Wolfe 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 Wolfe 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 $\scriptstyle \alpha ^{0}\,>\,0$ and choose some $\scriptstyle \tau \,\in \,(0,1).\,$

ii) Until $\scriptstyle \alpha _{j}\,$ satisfies the Wolfe condition:

$\alpha _{j+1}=\tau \alpha _{j},\,$

$j=j+1.\,$

iii) Return $\scriptstyle \alpha =\alpha _{j}.\,$

In other words, reduce $\scriptstyle \alpha ^{0}$ geometrically, with rate $\scriptstyle \tau \,$, until the Wolfe condition holds.

See also

• Line search

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534