Gradient-related

Gradient-related is a term used in multivariable calculus to describe a direction. 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 subsequence $\{d^k\}_{k \in K}$ is bounded and satisfies

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

A gradient-related direction is usually encountered in the gradient-based iterative optimisation of a function $f$ . At each iteration $k$ the current vector is $x^k$ and we move in the direction $d^k$ , thus generating a sequence of directions.

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.