Convex cones

(Difference between revisions)
 Revision as of 02:11, 14 June 2010 (edit)m (Protected "Convex cones" [edit=autoconfirmed:move=autoconfirmed])← Previous diff Current revision (12:33, 24 November 2011) (edit) (undo) Line 1: Line 1: - We call the set $\mathcal{K}_{\!}\subseteq_{\!}\reals^M$ a ''convex cone'' iff + We call the set $\mathcal{K}_{\!}\subseteq_{\!}\mathbb{R}^M$ a ''convex cone'' iff $\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 [itex]\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 Line 24: Line 24: (a.k.a: ''second-order cone'', ''quadratic cone'', ''circular cone'', ''Lorentz cone''), (a.k.a: ''second-order cone'', ''quadratic cone'', ''circular cone'', ''Lorentz cone''), - [itex]\mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\reals^n\!\times\reals + [itex]\mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\mathbb{R}^n\!\times\mathbb{R} ~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2$ ~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2[/itex] Line 30: Line 30: Esoteric examples of convex cones include Esoteric examples of convex cones include the point at the origin, any line through the origin, any ray having the origin as base the point at the origin, any line through the origin, any ray having the origin as base - such as the nonnegative real line $\reals_+$ in subspace $\reals\,$, + such as the nonnegative real line $\mathbb{R}_+$ in subspace $\mathbb{R}\,$, any halfspace partially bounded by a hyperplane through the origin, any halfspace partially bounded by a hyperplane through the origin, the positive semidefinite cone $\mathbb{S}_+^M$, the positive semidefinite cone $\mathbb{S}_+^M$, the cone of Euclidean distance matrices $\mathbb{EDM}^N$, the cone of Euclidean distance matrices $\mathbb{EDM}^N$, - any subspace, and Euclidean vector space $\reals^n$. + any subspace, and Euclidean vector space $\mathbb{R}^n$.

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 $LaTeX: \mathbb{R}^n$.