Projection on Polyhedral Cone

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We simply want a ''formula'' for projecting a given point in Euclidean space on a cone described by the intersection of an arbitrary number of halfspaces;<br> we want the closest point in the polyhedral cone.
We simply want a ''formula'' for projecting a given point in Euclidean space on a cone described by the intersection of an arbitrary number of halfspaces;<br> we want the closest point in the polyhedral cone.
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By "formula" I mean a closed form; an equation or set of equations (not a program or optimization).<br>
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By "formula" I mean a closed form; an equation or set of equations (not a program, algorithm, or optimization).<br>
A set of formulae, the choice of which is conditional, is OK
A set of formulae, the choice of which is conditional, is OK
as long as size of the set is not factorial (prohibitively large).
as long as size of the set is not factorial (prohibitively large).

Revision as of 01:04, 27 February 2009

This is an open problem in Convex Optimization. At first glance, it seems rather simple; the problem is certainly easily understood:

We simply want a formula for projecting a given point in Euclidean space on a cone described by the intersection of an arbitrary number of halfspaces;
we want the closest point in the polyhedral cone.

By "formula" I mean a closed form; an equation or set of equations (not a program, algorithm, or optimization).
A set of formulae, the choice of which is conditional, is OK as long as size of the set is not factorial (prohibitively large).

This problem has many practical and theoretical applications. Its solution is certainly worth a Ph.D. thesis in any Math or Engineering Department.

You are welcome and encouraged to write your thoughts about this problem here.

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