Gradient-related - Citizendia

A gradient-related direction is a term encountered in multivariable calculus. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives A gradient-related direction is usually encountered in the gradient-based iterative optimisation of a function $f$ . In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar At each iteration $k$ our current vector is $x k$ and we move in the direction $d k$ , thus generating a sequence of directions.

A direction sequence ${d k}$ is gradient related to ${x k}$ if:

For any subsequence $\{x^k\}_{k \in K}$ that converges to a nonstationary point, the corresponding subsequency $\{d^k\}_{k \in K}$ is bounded and satisfies

$\limsup_{k \rightarrow \infty, k \in K} \nabla f(x^k)'d^k <0$ .

It is easy to guarantee that the directions we generate are gradient related, by for example setting them equal to the gradient at each point.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |