A gradient-related direction is a term encounted in multivariable calculus. A gradient-related direction is usually encountered in the gradient-based iterative optimisation of a function $f$. 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 $\backslash \{d^k\backslash \}$ is gradient related to $\backslash \{x^k\backslash \}$ if:

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

.

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.