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 ∈ K that converges to a nonstationary point, the corresponding subsequence { d k } k ∈ K is bounded and satisfies

lim sup k → ∞ , k ∈ K ∇ 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.