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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 linesearch. 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 $\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k)<0$ . The motivation for such an approach is that small steps along $\mathbf{p}_k$ guarantee that $f$ is reduced, by Taylor's theorem.

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

Categories: Optimization

Last updated: 05-28-2005 11:02:39

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10-26-2009 08:16:03
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Descent direction