In optimization, a descent direction is a vector
{\displaystyle \mathbf {p} \in \mathbb {R} ^{n}}
that points towards a local minimum
of an objective function
Computing
by an iterative method, such as line search defines a descent direction
th iterate to be any
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, such as gradient descent or the conjugate gradient method.
More generally, if
is a positive definite matrix, then
is a descent direction at
[1] This generality is used in preconditioned gradient descent methods.