Gradient-related

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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.