Descent direction

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In optimization, a descent direction is a vector that, in the sense below, moves us closer towards a local minimum of our objective function .

Suppose we are computing by an iterative method, such as line search. We define a descent direction at the th iterate to be any such that , where denotes the inner product. The motivation for such an approach is that small steps along guarantee that is reduced, by Taylor's theorem.

Using this definition, the negative of a non-zero gradient is always a descent direction, as .

Numerous methods exist to compute descent directions, all with differing merits. For example, one could use gradient descent or the conjugate gradient method.

More generally, if is a positive definite matrix, then is a descent direction [1] at . This generality is used in preconditioned gradient descent methods.

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