|
|
| Line 1: |
Line 1: |
| - | We call the set <math>\mathcal{K}_{\!}\subseteq_{\!}\reals^M</math> a ''convex cone'' iff
| + | KiRecr <a href="http:// rgvflmafyjjc. com/ "> rgvflmafyjjc</ a>, [ url=http:// jwkmjcnjmsre. com/]jwkmjcnjmsre[ /url], [link= http:/ /csnagzwcxzqi. com/ ]csnagzwcxzqi[/ link], http:// ixelspdisfvg.com/ |
| - | | + | |
| - | <math>\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2
| + | |
| - | \in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0.</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>.
| + | |
| - | | + | |
| - | The set <math>\mathcal{K}</math> is convex since, for any particular <math>\zeta_{\,},\xi\geq0</math>,
| + | |
| - | | + | |
| - | <math>\mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]</math>
| + | |
| - | | + | |
| - | 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\reals^n\!\times\reals
| + | |
| - | ~|~\|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>\reals_+</math> in subspace <math>\reals\,</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>\reals^n</math>.
| + | |