Английская Википедия:Gradient-related

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

<math>\limsup_{k \rightarrow \infty, k \in K} \nabla f(x^k)'d^k <0.</math>

Gradient-related directions are usually encountered in the gradient-based iterative optimization of a function <math>f</math>. At each iteration <math>k</math> the current vector is <math>x^k</math> and we move in the direction <math>d^k</math>, 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.

Шаблон:Differential-geometry-stub