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 21:27, 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