Convex cones

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==Nonorthogonal projection on extreme directons of convex cone==
 
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===pseudo coordinates===
 
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Let <math>\mathcal{K}</math> be a full-dimensional closed pointed convex cone
 
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in finite-dimensional Euclidean space <math>\mathbb{R}^n</math>.
 
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For any vector <math>\,v\,</math> and a point <math>\,x\!\in\!\mathcal{K}</math>,
 
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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>.
 
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Suppose <math>\,x\,</math> and <math>\,y\,</math> are points in <math>\,\mathcal{K}\,</math>.
 
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Further, suppose that <math>\,d_v(x)\!=_{\!}d_v(y)\,</math> for each and every extreme direction <math>\,v_i\,</math> of <math>\,\mathcal{K}\,</math>.
 
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Then <math>\,x\,</math> must be equal to <math>\,y\,</math>.
 
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===proof===
 
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We construct an injectivity argument from vector <math>\,x\,</math> to the set <math>\,\{t_i^\star\}\,</math>
 
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where <math>\,t_i^\star\mathrel{\stackrel{\Delta}{=}}\,d_{v_i}(x)\,</math>.
 
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In other words, we assert that there is no <math>\,x\,</math> except <math>\,x\!=\!0\,</math> that nulls all the <math>\,t_i\,</math>;
 
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''i.e.'', there is no nullspace to operator <math>\,d\,</math> over all <math>\,v_i\,</math>.
 
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Function <math>\,d_v(x)\,</math> is the optimal objective value of a (primal) conic program:
 
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<math>\,\begin{array}{cl}\mathrm{maximize}_t&t\\
 
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\mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}\quad{\rm(p)}</math>
 
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Because dual geometry of this problem is easier to visualize,
 
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we instead interpret the dual conic program:
 
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<math>\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\
 
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\mathrm{subject~to}&\lambda\in\mathcal{K}^*\\
 
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&\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}</math>
 
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where <math>\,\mathcal{K}^*</math> is the dual cone, which is full-dimensional, closed, pointed, and convex because <math>\,\mathcal{K}\,</math> is.
 
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The primal optimal objective value equals the dual optimal value under the sufficient ''Slater condition'', which is well known;
 
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''i.e.'', we assume
 
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<math>\,t^\star=\,\lambda^{\star\rm T}x\,</math>
 

Revision as of 01:28, 29 August 2008

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