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

$\underset{k\to \mathrm{\infty },k\in K}{\mathrm{lim\; sup}}\mathrm{\nabla }f(x^k{)}^{\prime }{d}^k<0.$

Gradient-related directions are usually encountered in the gradient-based iterative optimization 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 generated are gradient-related: for example, they can be set equal to the gradient at each point.