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 $\varphi (\alpha )$ in the generic linesearch algorithm. One way to inexactly minimize $\varphi$ is by finding an ${\alpha }_k$ 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 $\alpha$ 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 $j=0$. Make an initial guess ${\alpha }^0>0$ and choose some $\tau \in (0,1).$

ii) Until ${\alpha }_j$ satisfies the Wolfe condition:

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

$j=j+1.$

iii) Return $\alpha ={\alpha }_j.$

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

See also

• Line search

References

• 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.
  
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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