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.