Projection on Polyhedral Cone

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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.


Projection onto isotone projection cones

Together with my coauthor A. B. Németh we recently discovered a very simple algorithm for projecting onto a special class of cones: the isotone projection cones.

Isotone projection cones are pointed closed convex cones LaTeX: \mathcal{K} for which projection LaTeX: P_\mathcal{K} onto the cone is isotone; that is, monotone with respect to the order LaTeX: \preceq induced by the cone: LaTeX: \forall\,x\,,\,y\in\mathbb{R}^n

LaTeX: x\preceq y\Rightarrow P_\mathcal{K}(x)\preceq P_\mathcal{K}(y),

or equivalently

LaTeX: y-x\in\mathcal K\Rightarrow P_\mathcal{K}(y)-P_\mathcal{K}(x)\in\mathcal K.

In LaTeX: \mathbb R^n, the isotone projection cones are polyhedral cones generated by LaTeX: n linearly independent vectors (i.e., cones that define a lattice structure; called latticial or simplicial cones) such that the generators of the polar cone form pairwise nonacute angles. For example, the nonnegative monotone cone (Example 2.13.9.4.2 in CO&EDG) is an isotone projection cone. The nonnegative monotone cone is defined by

LaTeX: \{\mathbf x=(x_1,\dots,x_n)^\top\in\mathbb R^n \mid x_1\geq\dots\geq x_n\geq 0\}.

Projecting onto nonnegative monotone cones (see Section 5.13.2.4 in CO&EDG) is important for the problem of map-making from relative distance information (see Section 5.13 in CO&EDG); e.g., stellar cartography. The isotone projection cones were introduced by George Isac and A. B. Németh in the mid-1980's and then applied by George Isac, A. B. Németh, and S. Z. Németh to complementarity problems. For simplicity we shall call a matrix LaTeX: \mathbf E , whose columns are generators of an isotone projection cone, an isotone projection cone generator matrix. Recall that an L-matrix is a matrix whose diagonal entries are positive and off-diagonal entries are nonpositive (see more at Z-matrix). A matrix LaTeX: \mathbf E is an isotone projection cone generator matrix if and only if it is of the form

LaTeX: \mathbf E=\mathbf O\mathbf T^{-\frac12},

where LaTeX: \mathbf O is an orthogonal matrix and LaTeX: \mathbf T is a positive definite L-matrix.

The algorithm is a finite one that stops in at most LaTeX: n steps and finds the exact projection point. Suppose that we want to project onto a latticial cone, and for each point in Euclidean space we know a proper face of the cone onto which that point projects. Then we could find the projection, recursively, by projecting onto linear subspaces of decreasing dimension. In case of isotone projection cones, the isotonicity property provides information required about such a proper face. The information is provided by geometrical structure of the polar cone.

If a polyhedral cone can be written as a union of isotone projection cones, reasonably small in number, then we could project a point onto the polyhedral cone by projecting onto the isotone projection cones and then taking the minimum distance of the given point from all these cones. Due to simplicity of the method for projecting onto an isotone projection cone, it is a challenging open question to find polyhedral cones that comprise a union of a small number of isotone projection cones that can be easily discerned. We conjecture that the latticial cones, which are subsets of the nonnegative orthant (or subsets of an isometric image of the nonnegative orthant), are such cones.

Scilab code can be downloaded here:

Matlab code for the algorithm:

% You are free to use, redistribute, and modifiy this code if you include,
% as a comment, the author of the original code
% (c) Sandor Zoltan Nemeth, 2009.
function p=piso(x,E)
[n,n]=size(E);
if det(E)==0, error('The input cone must be an isotone projection cone'); end
V=inv(E);
U=-V';
F=-V*U;
G=F-diag(diag(F));
for i=1:n
   for j=1:n
      if G(i,j)>0
         error('The input cone must be an isotone projection cone');
      end
   end
end
I=[1:n];
n1=n-1;
cont=1;
for k=0:n1
   [q1,l]=size(I);
   E1=E;
   I1=I;
   if l-1>=1
      highest=I(l);
      if highest<n
         for h=n:-1:highest+1
            E1(:,h)=zeros(n,1);
         end
      end
      for j=l-1:-1:1
         low=I(j)+1;
         high=I(j+1)-1;
         if high>=low
            for m=high:-1:low
               E1(:,m)=zeros(n,1);
            end
         end
         lowest=I(1);
         if lowest>1
            for w=lowest-1:-1:1
               E1(:,w)=zeros(n,1);
            end
         end
      end
   end
   if l==1
      E1=zeros(n,n);
      E1(:,I(1))=E(:,I(1));
   end

   V1=pinv(E1);
   U1=-V1';

   for j=l:-1:1
      zz=x'*U1(:,I(j));
      if zz>0, I1(j)=[];
      end
   end
   [q2,ll]=size(I1);
   if cont==0, p=x; return
   elseif ll==0, p=zeros(n,1); return
   else
      A=E1*V1;
      x=A*x;
      p=x;
   end
   if ll==l, cont=0;
   else cont=1;
   end
   I=I1;
end

LaTeX: -Sándor Zoltán Németh


external links

More about projection on cones (and convex sets, more generally) can be found here (chapter E):

http://meboo.convexoptimization.com/Meboo.html

More about polyhedral cones can be found in chapter 2.

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