Convex cones

(Difference between revisions)
 Revision as of 02:28, 29 August 2008 (edit) (Removing all content from page)← Previous diff Revision as of 18:33, 1 October 2008 (edit) (undo)Next diff → Line 1: Line 1: + We call the set $\mathcal{K}_{\!}\subseteq_{\!}\reals^M$ a ''convex cone'' iff + $\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 + \in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0$ + + Apparent from this definition, $\zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}}$ + and $\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}$ for all $\zeta_{\,},\xi_{\!}\geq_{\!}0_{}$. + The set $\mathcal{K}$ is convex since, for any particular $\zeta_{\,},\xi\geq0$ + + $\mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]$ + + because $\mu\,\zeta_{\,},(1-\mu)_{\,}\xi\geq0_{}$. + + Obviously, + + $\{\mathcal{X}\}\supset\{\mathcal{K}\}$ + + the set of all convex cones is a \emph{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. + Interior of a convex cone is possibly empty. + + + 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''), + + $\mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\reals^n\!\times\reals + ~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2$ + + 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 $\reals_+$ in subspace $\reals\,$, + any halfspace partially bounded by a hyperplane through the origin, + the positive semidefinite cone $\mathbb{S}_+^M$, + the cone of Euclidean distance matrices, + any subspace, and Euclidean vector space $\reals^n$.

Revision as of 18:33, 1 October 2008

We call the set $LaTeX: \mathcal{K}_{\!}\subseteq_{\!}\reals^M$ 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,

$LaTeX: \{\mathcal{X}\}\supset\{\mathcal{K}\}$

the set of all convex cones is a \emph{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. Interior of a convex cone is possibly empty.

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),

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

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 $\reals_+$ in subspace $LaTeX: \reals\,$, any halfspace partially bounded by a hyperplane through the origin, the positive semidefinite cone $LaTeX: \mathbb{S}_+^M$, the cone of Euclidean distance matrices, any subspace, and Euclidean vector space $\reals^n$.