Convex cones

From Wikimization

(Difference between revisions)
Jump to: navigation, search
Current revision (13:33, 24 November 2011) (edit) (undo)
 
(14 intermediate revisions not shown.)
Line 1: Line 1:
-
==Nonorthogonal projection on extreme directons of convex cone==
+
We call the set <math>\mathcal{K}_{\!}\subseteq_{\!}\mathbb{R}^M</math> a ''convex cone'' iff
-
===pseudo coordinates===
+
-
Let <math>\mathcal{K}</math> be a full-dimensional closed pointed convex cone
+
-
in finite-dimensional Euclidean space <math>\mathbb{R}^n</math>.
+
-
For any vector <math>\,v\,</math> and a point <math>\,x\!\in\!\mathcal{K}</math>,
+
<math>\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2
-
define <math>\,d_v(x)\,</math> to be the largest number <math>\,t^\star</math> such that <math>\,x-t^{}v\!\in\!\mathcal{K}\,</math>.
+
\in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0.</math>
-
Suppose <math>\,x\,</math> and <math>\,y\,</math> are points in <math>\,\mathcal{K}\,</math>.
+
Apparent from this definition, <math>\zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}}</math>
 +
and <math>\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}</math> for all <math>\zeta_{\,},\xi_{\!}\geq_{\!}0_{}</math>.
-
Further, suppose that <math>\,d_v(x)\!=_{\!}d_v(y)\,</math> for every extreme direction <math>\,v\,</math> of <math>\,\mathcal{K}\,</math>.
+
The set <math>\mathcal{K}</math> is convex since, for any particular <math>\zeta_{\,},\xi\geq0</math>,
-
Then <math>\,x\,</math> must be equal to <math>\,y\,</math>.
+
<math>\mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]</math>
-
===proof===
+
because <math>\mu\,\zeta_{\,},(1-\mu)_{\,}\xi\geq0_{}</math>.
 +
 
 +
Obviously, the set of all convex cones is a proper subset of all cones.
 +
 
 +
The set of convex cones is a narrower but more familiar class of cone, any member of which can be
 +
equivalently described as the intersection of a possibly (but not necessarily) infinite number of hyperplanes (through the origin)
 +
and halfspaces whose bounding hyperplanes pass through the origin; a halfspace-description.
 +
 
 +
Convex cones need not be full-dimensional.
 +
 
 +
Familiar examples of convex cones include an unbounded ''ice-cream cone'' united with its interior
 +
(a.k.a: ''second-order cone'', ''quadratic cone'', ''circular cone'', ''Lorentz cone''),
 +
 
 +
<math>\mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\mathbb{R}^n\!\times\mathbb{R}
 +
~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2</math>
 +
 
 +
and any polyhedral cone; ''e.g''., any orthant generated by Cartesian half-axes.
 +
Esoteric examples of convex cones include
 +
the point at the origin, any line through the origin, any ray having the origin as base
 +
such as the nonnegative real line <math>\mathbb{R}_+</math> in subspace <math>\mathbb{R}\,</math>,
 +
any halfspace partially bounded by a hyperplane through the origin,
 +
the positive semidefinite cone <math>\mathbb{S}_+^M</math>,
 +
the cone of Euclidean distance matrices <math>\mathbb{EDM}^N</math>,
 +
any subspace, and Euclidean vector space <math>\mathbb{R}^n</math>.

Current revision

We call the set LaTeX: \mathcal{K}_{\!}\subseteq_{\!}\mathbb{R}^M a convex cone iff

LaTeX: \Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 \in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0.

Apparent from this definition, LaTeX: \zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}} and LaTeX: \xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}} for all LaTeX: \zeta_{\,},\xi_{\!}\geq_{\!}0_{}.

The set LaTeX: \mathcal{K} is convex since, for any particular LaTeX: \zeta_{\,},\xi\geq0,

LaTeX: \mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]

because LaTeX: \mu\,\zeta_{\,},(1-\mu)_{\,}\xi\geq0_{}.

Obviously, the set of all convex cones is a proper subset of all cones.

The set of convex cones is a narrower but more familiar class of cone, any member of which can be equivalently described as the intersection of a possibly (but not necessarily) infinite number of hyperplanes (through the origin) and halfspaces whose bounding hyperplanes pass through the origin; a halfspace-description.

Convex cones need not be full-dimensional.

Familiar examples of convex cones include an unbounded ice-cream cone united with its interior (a.k.a: second-order cone, quadratic cone, circular cone, Lorentz cone),

LaTeX: \mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\mathbb{R}^n\!\times\mathbb{R} ~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2

and any polyhedral cone; e.g., any orthant generated by Cartesian half-axes. Esoteric examples of convex cones include the point at the origin, any line through the origin, any ray having the origin as base such as the nonnegative real line LaTeX: \mathbb{R}_+ in subspace LaTeX: \mathbb{R}\,, any halfspace partially bounded by a hyperplane through the origin, the positive semidefinite cone LaTeX: \mathbb{S}_+^M, the cone of Euclidean distance matrices LaTeX: \mathbb{EDM}^N, any subspace, and Euclidean vector space LaTeX: \mathbb{R}^n.

Retrieved from "http://www.convexoptimization.com/wikimization/index.php/Convex_cones"
Personal tools