# Gradient-related

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Gradient-related is a term used in multivariable calculus to describe a direction. A direction sequence ${\displaystyle \{d^{k}\}}$ is gradient-related to ${\displaystyle \{x^{k}\}}$ if for any subsequence ${\displaystyle \{x^{k}\}_{k\in K}}$ that converges to a nonstationary point, the corresponding subsequence ${\displaystyle \{d^{k}\}_{k\in K}}$ is bounded and satisfies

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

Gradient-related directions are usually encountered in the gradient-based iterative optimization of a function ${\displaystyle f}$. At each iteration ${\displaystyle k}$ the current vector is ${\displaystyle x^{k}}$ and we move in the direction ${\displaystyle d^{k}}$, 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.