# Convex cones

### From Wikimization

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===proof=== | ===proof=== | ||

We construct an injectivity argument from vector <math>\,x\,</math> to the set <math>\,\{t_i^\star\}\,</math> | We construct an injectivity argument from vector <math>\,x\,</math> to the set <math>\,\{t_i^\star\}\,</math> | ||

- | where <math>\,t_i^\star\ | + | where <math>\,t_i^\star\mathrel{\stackrel{\Delta}{=}}\,d_{v_i}(x)\,</math>. |

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>; | 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 | + | ''i.e.'', there is no nullspace to operator <math>\,d\,</math> over all <math>\,v_i\,</math>. |

- | <math>\,d_v(x)\,</math> is the optimal objective value of a (primal) conic program: | + | |

+ | Function <math>\,d_v(x)\,</math> is the optimal objective value of a (primal) conic program: | ||

<math>\,\begin{array}{cl}\mathrm{maximize}_t&t\\ | <math>\,\begin{array}{cl}\mathrm{maximize}_t&t\\ | ||

- | \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}</math> | + | \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}\quad{\rm(p)}</math> |

- | Because | + | Because dual geometry of this problem is easier to visualize, |

- | we instead interpret | + | we instead interpret its dual: |

<math>\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ | <math>\,\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}</math> | + | &\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}</math> |

where <math>\,\mathcal{K}^*\,</math> is the dual cone, which is full-dimensional, closed, pointed, and convex because <math>\,\mathcal{K}\,</math> is. | where <math>\,\mathcal{K}^*\,</math> is the dual cone, which is full-dimensional, closed, pointed, and convex because <math>\,\mathcal{K}\,</math> is. |

## Revision as of 21:05, 28 August 2008

## Nonorthogonal projection on extreme directons of convex cone

### pseudo coordinates

Let be a full-dimensional closed pointed convex cone in finite-dimensional Euclidean space .

For any vector and a point , define to be the largest number such that .

Suppose and are points in .

Further, suppose that for each and every extreme direction of .

Then must be equal to .

### proof

We construct an injectivity argument from vector to the set where .

In other words, we assert that there is no except that nulls all the ;
*i.e.*, there is no nullspace to operator over all .

Function is the optimal objective value of a (primal) conic program:

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

where is the dual cone, which is full-dimensional, closed, pointed, and convex because is.

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

*i.e.*, we assume