Money flow index

2013-12-19T18:26:26Z

151.151.109.24: added lookup of money flow index patterns

In [[optimization (mathematics)|optimization]], a '''descent direction''' is a vector <math>\mathbf{p}\in\mathbb R^n</math> that, in the sense below, moves us closer towards a local minimum <math>\mathbf{x}^*</math> of our objective function <math>f:\mathbb R^n\to\mathbb R</math>.

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

Using this definition, the negative of a non-zero gradient is always a
descent direction, as <math> \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 </math>.

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 <math>P</math> is a [[positive definite]] matrix, then
<math>d = -P \nabla f(x)</math>
is a descent direction
<ref name="?">{{cite book | author = J. M. Ortega and W. C. Rheinbold | title = Iterative Solution of Nonlinear Equations in Several Variables | pages = 243 | year = 1970 | doi = 10.1137/1.9780898719468
}}</ref>
at <math>x</math>.
This generality is used in [[preconditioned]] gradient descent methods.

{{DEFAULTSORT:Descent Direction}}
[[Category:Mathematical optimization]]

{{Reflist}}

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Money flow index