Descent direction

In optimization, a descent direction is a vector $\mathbf {p} \in \mathbb {R} ^{n}$ that, in the sense below, moves us closer towards a local minimum $\mathbf{x}^*$ of our objective function $f:\mathbb {R} ^{n}\to \mathbb {R}$ .

Suppose we are computing $\mathbf{x}^*$ by an iterative method, such as line search. We define a descent direction $\mathbf {p} _{k}\in \mathbb {R} ^{n}$ at the $k$ th iterate to be any $\mathbf {p} _{k}$ such that $\langle \mathbf {p} _{k},\nabla f(\mathbf {x} _{k})\rangle <0$ , where $\langle ,\rangle$ denotes the inner product. The motivation for such an approach is that small steps along $\mathbf {p} _{k}$ guarantee that $\displaystyle f$ is reduced, by Taylor's theorem.

Using this definition, the negative of a non-zero gradient is always a descent direction, as $\langle -\nabla f(\mathbf {x} _{k}),\nabla f(\mathbf {x} _{k})\rangle =-\langle \nabla f(\mathbf {x} _{k}),\nabla f(\mathbf {x} _{k})\rangle <0$ .

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 $P$ is a positive definite matrix, then $d=-P\nabla f(x)$ is a descent direction ^[1] at $x$ . This generality is used in preconditioned gradient descent methods.

↑ J. M. Ortega and W. C. Rheinbold (1970). Iterative Solution of Nonlinear Equations in Several Variables. p. 243. doi:10.1137/1.9780898719468.

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