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.

单词	Descent direction
释义	Descent direction 英语百科 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.
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