# Convex cones

### From Wikimization

Line 9: | Line 9: | ||

Suppose <math>\,x\,</math> and <math>\,y\,</math> are points in <math>\,\mathcal{K}\,</math>. | Suppose <math>\,x\,</math> and <math>\,y\,</math> are points in <math>\,\mathcal{K}\,</math>. | ||

- | Further, suppose that <math>\,d_v(x)\!=_{\!}d_v(y)\,</math> for every extreme direction <math>\, | + | 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>. |

Then <math>\,x\,</math> must be equal to <math>\,y\,</math>. | Then <math>\,x\,</math> must be equal to <math>\,y\,</math>. | ||

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

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

+ | where <math>\,t_i^\star\!=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>; | ||

+ | ''i.e.'', there is no nullspace to the operation. | ||

+ | |||

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

## Revision as of 20:39, 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 the operation.

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

Because the dual geometry of this problem is easier to visualize, we instead interpret the dual conic program:

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