# Talk:Projection on Polyhedral Cone

The definition of projection should be made clear. If, by projection you mean the nearest point to the cone, then this results in a quadratic programming problem, i.e. given the point $LaTeX: \bar x$ and the cone $LaTeX: K=\left\{x: Ax \leq b \right\}$, then the quadratic program is $LaTeX: \begin{array}{rcl} \min & \|x-\bar x\|^2_2 \\ \mbox{s.t.} & x \in K \end{array}$

No explicit formula for a solution of a quadratic program exists (or for the solution of a linear program). It is doubtful that such a formula will be found due to the combinatorial nature of the problem.

## Contents

Yes, the nearest point belonging to the cone.

I have been asked this question several times; i.e., is there a formula for projecting on a polyhedral cone. We know some formulas; e.g., projection on the positive semidefinite cone, or on the various orthants. People are surprised when they discover there are no formulae for projection on simple polyhedral cones, because it seems like there ought to be.

I think it remains a fascinating question. Perhaps we could collect here what we do know about this problem. --Dattorro 13:49, 9 June 2008 (PDT)

## Relevant Reference

An excellent source of results on subspaces and projectors is

author="A. Galantai",
title="{Subspaces, Angles and Pairs of Orthogonal Projections}",
journal="{L}in. Multilin. {A}lg.",
year="2008",
volume="56",
pages="227--260"


## Spherical

Why have you only mentioned the polyhedral cone? Is there even a formula to project a point on spherical cone? --fisher 10:57, 28 Feb 2009 (GMT)

Because it is simpler. --Dattorro 19:28, 27 February 2009 (PST)

## Second-order cone

How about second-order cone (or revolution cone), if I want to project a point on it, is second-order cone program (SOCP) the only choice? --fisher 13:47, 28 Feb 2009 (GMT)