Gradient-related

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

[math]\displaystyle{ \limsup_{k \rightarrow \infty, k \in K} \nabla f(x^k)'d^k \lt 0. }[/math]

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

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