# Convex cones

### From Wikimization

(Difference between revisions)

Line 14: | Line 14: | ||

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

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

- | <math>\,\begin{array}{cl}\mathrm{maximize}&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}</math> | ||

Line 26: | Line 26: | ||

&\lambda^{\rm T}v=1\end{array}</math> | &\lambda^{\rm T}v=1\end{array}</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. |

+ | |||

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

+ | |||

+ | ''i.e.'', we assume | ||

+ | |||

+ | <math>\,t^\star=\,\lambda^{\star\rm T}x\,</math> |

## Revision as of 20:27, 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 every extreme direction of .

Then must be equal to .

### proof

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