Descent direction

El directorio enciclopédico desde la Wikipedia.

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 $\langle\mathbf{p}_k,\nabla f(\mathbf{x}_k)\rangle < 0$ , 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.

Página espejo de la Wikipedia

Directorio de Enlaces Directorio dmoz Directorio espejo dmoz Pedro Bernardo