Gradient-related

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A gradient-related direction is a term encountered 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 xk and we move in the direction dk, thus generating a sequence of directions.

A direction sequence {dk} is gradient related to {xk} if:

For any subsequence \{x^k\}_{k \in K} that converges to a nonstationary point, the corresponding subsequency \{d^k\}_{k \in K} is bounded and satisfies
\limsup_{k \rightarrow \infty, k \in K} \nabla f(x^k)'d^k <0.

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.

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