# Projection on Polyhedral Cone

### From Wikimization

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.

## Contents |

## Projection on isotone projection cones

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

An isotone projection cone is a generating pointed closed convex cone in a Hilbert space for which projection onto the cone is isotone; that is, monotone with respect to the order induced by the cone:

or equivalently

From now on, suppose that we are in . Here the isotone projection cones are polyhedral cones generated by 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

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 (see 1, 2, 3 and 4). The algorithm below opens the way for efficient implementations of the iterative methods in 1, 2 and 4. For simplicity we shall call a matrix 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. A matrix is an isotone projection cone generator matrix if and only if it is of the form

where is an orthogonal matrix and is a positive definite L-matrix.

The algorithm is a finite one that stops in at most 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. The theoretical background for the algorithm is Moreau's decomposition theorem.

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.

The results are contained in a paper submitted to *Linear Algebra and its Applications*. The paper received a very positive review and it is accepted subject to some modifications.

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

Sándor Zoltán Németh

## Fast projection on monotone nonnegative cone

Demo requires CVX to produce estimate of error by solving the projection problem in polynomial time.

CVX is not required for piso4() which performs projection by Sandor's method about 100 times faster.

%demo: Sandor's projection on monotone nonnegative cone %-Jon Dattorro, August 2, 2009 clear all n = 100; cvx_quiet('true') cvx_precision('high') a = randn(n,1); tic t = piso4(a); fprintf('\n') toc if n < 500 tic cvx_begin variable s(n,1); variable b(n,1); minimize(norm(s-a)); for i=1:n s(i) == sum(b(i:n)); end b >= 0; cvx_end toc err = sum(abs(s - t)) end

### piso4()

This code is optimized for memory efficiency. Otherwise, for millions of variables, required memory would be measured in terabytes.

% -Sandor Zoltan Nemeth, with code optimization -Jon Dattorro August 2, 2009. % Project x on monotone nonnegative cone of dimension n. function p = piso4(x) n = length(x); I = 1:n; disp('_____') for k=1:n fprintf('\b\b\b\b\b\b%6d',k); L = length(I); zeroidx = 1:n; zeroidx(I) = []; V1 = getpinv2(zeroidx,n); I(find(V1(I,:)*x<0)) = []; newL = length(I); if ~newL p = sparse(n,1); return else t = V1*x; s = sum(t); zs = sum(t(zeroidx)); for i=1:n x(i) = s - zs; s = s - t(i); first = find(zeroidx>=i,1); zs = zs - sum(t(zeroidx(1:first-1))); zeroidx(1:first-1) = []; end end if newL==L p = x; return end end

### getpinv2()

%Assumes Platonic upper triangular ones (generator matrix) with some columns missing. %Quickly finds pseudoinverse of monotone nonnegative cone generator matrix %June 14, 2009 -Jon Dattorro function Y = getpinv2(zeroidx,n); Y = spdiags([ones(n,1), -ones(n,1)], [0 1], sparse(n,n)); Y(zeroidx,:) = 0; %find dangling -1 count = 0; for i=1:n if ~Y(i,i) count = count + 1; if i-1 > 0 Y(i-1,i) = 0; end else if count if i-count-1 > 0 Y(i-count-1,i-count:i) = -1/(count+1); end Y(i,i-count:i) = 1/(count+1); end count = 0; end end

## Projection on simplicial cones

Together with my coauthors A. Ekárt and A. B. Németh we recently discovered a very fast heuristic iterative method of projection on *simplicial cones*. It consists in solving two linear systems at each step of the iteration. The extensive experiments indicate that the method furnishes the exact solution in more then 99.7 percent of the cases. The average number of steps is 5.67 (we have not found any examples which required more than 13 steps) and the relative number of steps with respect to the dimension decreases dramatically. Roughly speaking, for high enough dimensions the absolute number of steps is independent of the dimension. The percentage of problems where the algorithm was aborted due to going in a loop exponentially decreases as the size increases. The percentage of loops above sizes of 30 is 0.001%. Overall, loops were observed in 0.3% of the experiments, so the heuristic algorithm was successful 99.7% of the time. Most of the loops were observed for problems of very small size (2, 3 and 5). However, our algorithm work extremely well for problems of large size. The conclusions are based on millions of numerical experiments. An extensive statistical analysis was performed with several statistics tables and figures.

Scilab code can be downloaded here:

## 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.