# Convex cones

(Difference between revisions)
 Revision as of 20:39, 28 August 2008 (edit)← Previous diff Revision as of 21:05, 28 August 2008 (edit) (undo)Next diff → Line 15: Line 15: ===proof=== ===proof=== We construct an injectivity argument from vector $\,x\,$ to the set $\,\{t_i^\star\}\,$ We construct an injectivity argument from vector $\,x\,$ to the set $\,\{t_i^\star\}\,$ - where $\,t_i^\star\!=d_{v_i}(x)\,$. + 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 $\,t_i\,$; 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 the operation. + ''i.e.'', there is no nullspace to operator $\,d\,$ over all $\,v_i\,$. - $\,d_v(x)\,$ is the optimal objective value of a (primal) conic program: + + Function $\,d_v(x)\,$ is the optimal objective value of a (primal) conic program: $\,\begin{array}{cl}\mathrm{maximize}_t&t\\ [itex]\,\begin{array}{cl}\mathrm{maximize}_t&t\\ - \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}$ + \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}\quad{\rm(p)}[/itex] - Because the dual geometry of this problem is easier to visualize, + Because dual geometry of this problem is easier to visualize, - we instead interpret the dual conic program: + we instead interpret its dual: $\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ [itex]\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ \mathrm{subject~to}&\lambda\in\mathcal{K}^*\\ \mathrm{subject~to}&\lambda\in\mathcal{K}^*\\ - &\lambda^{\rm T}v=1\end{array}$ + &\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}[/itex] where $\,\mathcal{K}^*\,$ is the dual cone, which is full-dimensional, closed, pointed, and convex because $\,\mathcal{K}\,$ is. where $\,\mathcal{K}^*\,$ is the dual cone, which is full-dimensional, closed, pointed, and convex because $\,\mathcal{K}\,$ is.

## Nonorthogonal projection on extreme directons of convex cone

### pseudo coordinates

Let $LaTeX: \mathcal{K}$ be a full-dimensional closed pointed convex cone in finite-dimensional Euclidean space $LaTeX: \mathbb{R}^n$.

For any vector $LaTeX: \,v\,$ and a point $LaTeX: \,x\!\in\!\mathcal{K}$, define $LaTeX: \,d_v(x)\,$ to be the largest number $LaTeX: \,t^\star$ such that $LaTeX: \,x-t^{}v\!\in\!\mathcal{K}\,$.

Suppose $LaTeX: \,x\,$ and $LaTeX: \,y\,$ are points in $LaTeX: \,\mathcal{K}\,$.

Further, suppose that $LaTeX: \,d_v(x)\!=_{\!}d_v(y)\,$ for each and every extreme direction $LaTeX: \,v_i\,$ of $LaTeX: \,\mathcal{K}\,$.

Then $LaTeX: \,x\,$ must be equal to $LaTeX: \,y\,$.

### proof

We construct an injectivity argument from vector $LaTeX: \,x\,$ to the set $LaTeX: \,\{t_i^\star\}\,$ where $LaTeX: \,t_i^\star\mathrel{\stackrel{\Delta}{=}}\,d_{v_i}(x)\,$.

In other words, we assert that there is no $LaTeX: \,x\,$ except $LaTeX: \,x\!=\!0\,$ that nulls all the $LaTeX: \,t_i\,$; i.e., there is no nullspace to operator $LaTeX: \,d\,$ over all $LaTeX: \,v_i\,$.

Function $LaTeX: \,d_v(x)\,$ is the optimal objective value of a (primal) conic program:

$LaTeX: \,\begin{array}{cl}\mathrm{maximize}_t&t\\ \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}\quad{\rm(p)}$

Because dual geometry of this problem is easier to visualize, we instead interpret its dual:

$LaTeX: \,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ \mathrm{subject~to}&\lambda\in\mathcal{K}^*\\ &\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}$

where $LaTeX: \,\mathcal{K}^*\,$ is the dual cone, which is full-dimensional, closed, pointed, and convex because $LaTeX: \,\mathcal{K}\,$ is.

The primal optimal objective value equals the dual optimal value under the sufficient Slater condition, which is well known;

i.e., we assume

$LaTeX: \,t^\star=\,\lambda^{\star\rm T}x\,$