Gradient-related

Gradient-related

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 x^k and we move in the direction d^k, thus generating a sequence of directions.

A direction sequence {d^k} is "gradient related" to {x^k} 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 ightarrow infty, k in K} abla 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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