Convex cones
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- | ==Nonorthogonal projection on extreme directons of convex cone== | ||
- | ===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>, | ||
- | 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=== | ||
- | We construct an injectivity argument from vector <math>\,x\,</math> to the set <math>\,\{t_i^\star\}\,</math> | ||
- | 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>; | ||
- | ''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\\ | ||
- | \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, | ||
- | we instead interpret the dual conic program: | ||
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- | <math>\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ | ||
- | \mathrm{subject~to}&\lambda\in\mathcal{K}^*\\ | ||
- | &\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> |