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 .