Gradient-related
Gradient-related is a term used in multivariable calculus to describe a direction. A direction sequence [math]\displaystyle{ \{d^k\} }[/math] is gradient-related to [math]\displaystyle{ \{x^k\} }[/math] if for any subsequence [math]\displaystyle{ \{x^k\}_{k \in K} }[/math] that converges to a nonstationary point, the corresponding subsequence [math]\displaystyle{ \{d^k\}_{k \in K} }[/math] is bounded and satisfies
[math]\displaystyle{ \limsup_{k \rightarrow \infty, k \in K} \nabla f(x^k)'d^k \lt 0. }[/math]
Gradient-related directions are usually encountered in the gradient-based iterative optimization of a function [math]\displaystyle{ f }[/math]. At each iteration [math]\displaystyle{ k }[/math] the current vector is [math]\displaystyle{ x^k }[/math] and we move in the direction [math]\displaystyle{ d^k }[/math], thus generating a sequence of directions.
It is easy to guarantee that the directions generated are gradient-related: for example, they can be set equal to the gradient at each point.
This article does not cite any external source. HandWiki requires at least one external source. See citing external sources. (2021) (Learn how and when to remove this template message) |
Original source: https://en.wikipedia.org/wiki/Gradient-related.
Read more |