Convex cones
From Wikimization
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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 21:39, 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