# Convex cones

(Difference between revisions)
 Revision as of 21:07, 28 August 2008 (edit)← Previous diff Revision as of 01:28, 29 August 2008 (edit) (undo) (Removing all content from page)Next diff → Line 1: Line 1: - ==Nonorthogonal projection on extreme directons of convex cone== - ===pseudo coordinates=== - Let $\mathcal{K}$ be a full-dimensional closed pointed convex cone - in finite-dimensional Euclidean space $\mathbb{R}^n$. - For any vector $\,v\,$ and a point $\,x\!\in\!\mathcal{K}$, - define $\,d_v(x)\,$ to be the largest number $\,t^\star$ such that $\,x-t^{}v\!\in\!\mathcal{K}\,$. - - Suppose $\,x\,$ and $\,y\,$ are points in $\,\mathcal{K}\,$. - - Further, suppose that $\,d_v(x)\!=_{\!}d_v(y)\,$ for each and every extreme direction $\,v_i\,$ of $\,\mathcal{K}\,$. - - Then $\,x\,$ must be equal to $\,y\,$. - - ===proof=== - We construct an injectivity argument from vector $\,x\,$ to the set $\,\{t_i^\star\}\,$ - where $\,t_i^\star\mathrel{\stackrel{\Delta}{=}}\,d_{v_i}(x)\,$. - - In other words, we assert that there is no $\,x\,$ except $\,x\!=\!0\,$ that nulls all the $\,t_i\,$; - ''i.e.'', there is no nullspace to operator $\,d\,$ over all $\,v_i\,$. - - - Function $\,d_v(x)\,$ is the optimal objective value of a (primal) conic program: - - $\,\begin{array}{cl}\mathrm{maximize}_t&t\\ - \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}\quad{\rm(p)}$ - - Because dual geometry of this problem is easier to visualize, - we instead interpret the dual conic program: - - $\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ - \mathrm{subject~to}&\lambda\in\mathcal{K}^*\\ - &\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}$ - - where $\,\mathcal{K}^*$ is the dual cone, which is full-dimensional, closed, pointed, and convex because $\,\mathcal{K}\,$ is. - - The primal optimal objective value equals the dual optimal value under the sufficient ''Slater condition'', which is well known; - - ''i.e.'', we assume - - $\,t^\star=\,\lambda^{\star\rm T}x\,$