Convex cones

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===proof===
===proof===
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<math>\,d_v(x)\,</math> is the optimal value of a conic program:
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<math>\,\begin{array}{rl}\mathrm{maximize}&t\\
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\mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}</math>

Revision as of 21:02, 28 August 2008

Nonorthogonal projection on extreme directons of convex cone

pseudo coordinates

Let LaTeX: \mathcal{K} be a full-dimensional closed pointed convex cone in finite-dimensional Euclidean space LaTeX: \mathbb{R}^n.

For any vector LaTeX: \,v\, and a point LaTeX: \,x\!\in\!\mathcal{K}, define LaTeX: \,d_v(x)\, to be the largest number LaTeX: \,t^\star such that LaTeX: \,x-t^{}v\!\in\!\mathcal{K}\,.

Suppose LaTeX: \,x\, and LaTeX: \,y\, are points in LaTeX: \,\mathcal{K}\,.

Further, suppose that LaTeX: \,d_v(x)\!=_{\!}d_v(y)\, for every extreme direction LaTeX: \,v\, of LaTeX: \,\mathcal{K}\,.

Then LaTeX: \,x\, must be equal to LaTeX: \,y\,.

proof

LaTeX: \,d_v(x)\, is the optimal value of a conic program:

LaTeX: \,\begin{array}{rl}\mathrm{maximize}&t\\ \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}

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