# Convex cones

### From Wikimization

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

- | we instead interpret | + | we instead interpret the dual conic program: |

<math>\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ | <math>\,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ | ||

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&\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}</math> | &\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}</math> | ||

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

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

## Revision as of 21:07, 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 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