# Descent direction

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.

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}