Gradient-related

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Gradient-related is a term used in multivariable calculus to describe a direction. A direction sequence { d k } {\displaystyle \\{d^{k}\\}} is gradient-related to  { x k } {\displaystyle \\{x^{k}\\}} if for any subsequence { x k } k ∈ K {\displaystyle \\{x^{k}\\}_{k\in K}} that converges to a nonstationary point, the corresponding subsequence  { d k } k ∈ K {\displaystyle \\{d^{k}\\}_{k\in K}} is bounded and satisfies

lim sup k -> ∞ , k ∈ K ∇ f ( x k ) ′ d k < 0. {\displaystyle \limsup _{k\rightarrow \infty ,k\in K}\nabla f(x^{k})'d^{k}<0.}

Gradient-related directions are usually encountered in the gradient-based iterative optimization of a function f {\displaystyle f} . At each iteration  k {\displaystyle k} the current vector is  x k {\displaystyle x^{k}} and we move in the direction  d k {\displaystyle 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.

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