# Matlab for Convex Optimization & Euclidean Distance Geometry

(Difference between revisions)
 Revision as of 17:53, 30 July 2008 (edit) (New page: Made by The MathWorks [http://www.mathworks.com http://www.mathworks.com], Matlab is a high level programming language and graphical user interface for linear algebra. == isedm() ==
...)← Previous diff                                                                                                                                                         Revision as of 18:03, 30 July 2008 (edit) (undo)Next diff →
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V = [eye(n)-ones(n,n)/n];                                                                                                                                                                                           V = [eye(n)-ones(n,n)/n];
-                                                                                                                                                                                                                                                                     \end{verbatim}                                                                                                                                         +
- \pagebreak + == signeig() == - \subsubsection{{\tt signeig()}\index{signeig()}} +
-                                                                                                                                                                                                                                                                     \begin{verbatim}                                                                                                                                       +
% Sorts signed real part of eigenvalues                                                                                                                                                                             % Sorts signed real part of eigenvalues
% and applies sort to values and vectors.                                                                                                                                                                           % and applies sort to values and vectors.
Line 188:                                                                                                                                                                                                                                                                                                                                                                                                                    Line 187:
Q = diag(lam);                                                                                                                                                                                                      Q = diag(lam);
end                                                                                                                                                                                                                 end
+
- \end{verbatim} + == Dx() == - +
-                                                                                                                                                                                                                                                                     \subsubsection{{\tt Dx()}\index{Dx()}}                                                                                                                 +
-                                                                                                                                                                                                                                                                     \begin{verbatim}                                                                                                                                       +
% Make EDM from point list                                                                                                                                                                                          % Make EDM from point list
function D = Dx(X)                                                                                                                                                                                                  function D = Dx(X)
Line 201:                                                                                                                                                                                                                                                                                                                                                                                                                    Line 199:
del = diag(X'*X);                                                                                                                                                                                                   del = diag(X'*X);
D = del*one' + one*del' - 2*X'*X;                                                                                                                                                                                   D = del*one' + one*del' - 2*X'*X;
-                                                                                                                                                                                                                                                                     \end{verbatim}                                                                                                                                         +
- \newpage + == conic independence == - \section{conic independence, {\tt conici()}\index{conici()}}\label{conici} + The recommended subroutine lp() is a linear program solver (''simplex method'') - (\S\ref{conicindy}) The recommended subroutine {\tt $_{_{}}$lp()} (\S\ref{linp}) is a linear program solver \textbf{(}\emph{simplex method}\textbf{)} + from Matlab's ''Optimization Toolbox'' v2.0 (R11). - from {\sc Matlab}'s \emph{Optimization Toolbox} v2.0 (R11).\index{Matlab!lp() \emph{versus} linprog()} + Later releases of Matlab replace lp() with linprog() (interior-point method) - Later releases of {\sc Matlab} replace {\tt $_{_{}}$lp()} with {\tt $_{_{}}$linprog()} \textbf{(}interior-point method\textbf{)} + that we find quite inferior to lp() on an assortment of problems; - that we find quite inferior to {\tt $_{_{}}$lp()} on an assortment of problems; + indeed, inherent limitation of numerical precision of solution to 1E-8 in - indeed, inherent limitation of numerical precision of solution to {\tt 1E-8} in\index{precision!numerical}\index{accuracy} + linprog() causes failure in programs previously working with lp(). - {\tt $_{_{}}$linprog()} causes failure in programs previously working with {\tt $_{_{}}$lp()$_{}$}. + - Given an arbitrary set of directions, this $\;$c.i. subroutine removes the conically dependent members. + Given an arbitrary set of directions, this c.i. subroutine removes the conically dependent members. Yet a conically independent set returned Yet a conically independent set returned is not necessarily unique. In that case, if desired, the set returned may be altered by reordering the set input. is not necessarily unique. In that case, if desired, the set returned may be altered by reordering the set input. - \begin{verbatim} +
% Test for c.i. of arbitrary directions in rows or columns of X.                                                                                                                                                    % Test for c.i. of arbitrary directions in rows or columns of X.
% -Jon Dattorro                                                                                                                                                                                                     % -Jon Dattorro
Line 243:                                                                                                                                                                                                                                                                                                                                                                                                                    Line 240:
return                                                                                                                                                                                                              return
end                                                                                                                                                                                                                 end
-                                                                                                                                                                                                                                                                     \end{verbatim}

-                                                                                                                                                                                                                                                                     \begin{verbatim}
count = 1;                                                                                                                                                                                                          count = 1;
new_N = N;                                                                                                                                                                                                          new_N = N;
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Xci = Xin';                                                                                                                                                                                                         Xci = Xin';
end                                                                                                                                                                                                                 end
+
- \end{verbatim} - + == lp() == - \newpage +
-                                                                                                                                                                                                                                                                     \subsection{{\tt lp()}\index{lp()}\index{Lagrangian}}\label{linp}                                                                                      +
-                                                                                                                                                                                                                                                                     \begin{verbatim}                                                                                                                                       +
LP     Linear programming.                                                                                                                                                                                          LP     Linear programming.
X=LP(f,A,b) solves the linear programming problem:                                                                                                                                                                  X=LP(f,A,b) solves the linear programming problem:
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LP produces warning messages when the solution is either                                                                                                                                                            LP produces warning messages when the solution is either
unbounded or infeasible.                                                                                                                                                                                            unbounded or infeasible.
-                                                                                                                                                                                                                                                                     \end{verbatim}                                                                                                                                         +
- + - + - + - \newpage + == Map of the USA == - \section{Map of the USA\index{map!USA}} + === EDM, mapusa() === - \subsection{EDM, {\tt mapusa()}\index{mapusa()}}\label{mouse} + (\S\ref{USA}) (\S\ref{USA}) \begin{verbatim} \begin{verbatim}

## Revision as of 18:03, 30 July 2008

Made by The MathWorks http://www.mathworks.com, Matlab is a high level programming language and graphical user interface for linear algebra.

## isedm()

% Is real D a Euclidean Distance Matrix. -Jon Dattorro
%
% [Dclosest,X,isisnot,r] = isedm(D,tolerance,verbose,dimension,V)
%
% Returns: closest EDM in Schoenberg sense (default output),
%          a generating list X,
%          string 'is' or 'isnot' EDM,
%          actual affine dimension r of EDM output.
% Input: matrix D,
%        optional absolute numerical tolerance for EDM determination,
%        optional verbosity 'on' or 'off',
%        optional desired affine dim of generating list X output,
%        optional choice of 'Vn' auxiliary matrix (default) or 'V'.

function [Dclosest,X,isisnot,r] = isedm(D,tolerance_in,verbose,dim,V);

isisnot = 'is';
N = length(D);

if nargin < 2 | isempty(tolerance_in)
tolerance_in = eps;
end
tolerance = max(tolerance_in, eps*N*norm(D));
if nargin < 3 | isempty(verbose)
verbose = 'on';
end
if nargin < 5 | isempty(V)
use = 'Vn';
else
use = 'V';
end

% is empty
if N < 1
if strcmp(verbose,'on'), disp('Input D is empty.'), end
X = [ ];
Dclosest = [ ];
isisnot = 'isnot';
r = [ ];
return
end
% is square
if size(D,1) ~= size(D,2)
if strcmp(verbose,'on'), disp('An EDM must be square.'), end
X = [ ];
Dclosest = [ ];
isisnot = 'isnot';
r = [ ];
return
end
% is real
if ~isreal(D)
if strcmp(verbose,'on'), disp('Because an EDM is real,'), end
isisnot = 'isnot';
D = real(D);
end

% is nonnegative
if sum(sum(chop(D,tolerance) < 0))
isisnot = 'isnot';
if strcmp(verbose,'on'), disp('Because an EDM is nonnegative,'),end
end
% is symmetric
if sum(sum(abs(chop((D - D')/2,tolerance)) > 0))
isisnot = 'isnot';
if strcmp(verbose,'on'), disp('Because an EDM is symmetric,'), end
D = (D + D')/2;  % only required condition
end
% has zero diagonal
if sum(abs(diag(chop(D,tolerance))) > 0)
isisnot = 'isnot';
if strcmp(verbose,'on')
disp('Because an EDM has zero main diagonal,')
end
end
% is EDM
if strcmp(use,'Vn')
VDV = -Vn(N)'*D*Vn(N);
else
VDV = -Vm(N)'*D*Vm(N);
end
[Evecs Evals] = signeig(VDV);
if ~isempty(find(chop(diag(Evals),...
max(tolerance_in,eps*N*normest(VDV))) < 0))
isisnot = 'isnot';
if strcmp(verbose,'on'), disp('Because -VDV < 0,'), end
end
if strcmp(verbose,'on')
if strcmp(isisnot,'isnot')
disp('matrix input is not EDM.')
elseif tolerance_in == eps
disp('Matrix input is EDM to machine precision.')
else
disp('Matrix input is EDM to specified tolerance.')
end
end

% find generating list
r = max(find(chop(diag(Evals),...
max(tolerance_in,eps*N*normest(VDV))) > 0));
if isempty(r)
r = 0;
end
if nargin < 4 | isempty(dim)
dim = r;
else
dim = round(dim);
end
t = r;
r = min(r,dim);
if r == 0
X = zeros(1,N);
else
if strcmp(use,'Vn')
X = [zeros(r,1) diag(sqrt(diag(Evals(1:r,1:r))))*Evecs(:,1:r)'];
else
X = [diag(sqrt(diag(Evals(1:r,1:r))))*Evecs(:,1:r)']/sqrt(2);
end
end
if strcmp(isisnot,'isnot') | dim < t
Dclosest = Dx(X);
else
Dclosest = D;
end


Subroutines for isedm()

## chop()

% zeroing entries below specified absolute tolerance threshold
% -Jon Dattorro
function Y = chop(A,tolerance)

R = real(A);
I = imag(A);

if nargin == 1
tolerance = max(size(A))*norm(A)*eps;
end
idR = find(abs(R) < tolerance);
idI = find(abs(I) < tolerance);

R(idR) = 0;
I(idI) = 0;

Y = R + i*I;


## Vn()

function y = Vn(N)

y = [-ones(1,N-1);
eye(N-1)]/sqrt(2);


## Vm()

% returns EDM V matrix
function V = Vm(n)

V = [eye(n)-ones(n,n)/n];


## signeig()

% Sorts signed real part of eigenvalues
% and applies sort to values and vectors.
% [Q, lam] = signeig(A)
% -Jon Dattorro

function [Q, lam] = signeig(A);

[q l] = eig(A);

lam = diag(l);
[junk id] = sort(real(lam));
id = id(length(id):-1:1);
lam = diag(lam(id));
Q = q(:,id);

if nargout < 2
Q = diag(lam);
end


## Dx()

% Make EDM from point list
function D = Dx(X)

[n,N] = size(X);
one = ones(N,1);

del = diag(X'*X);
D = del*one' + one*del' - 2*X'*X;


## conic independence

The recommended subroutine lp() is a linear program solver (simplex method) from Matlab's Optimization Toolbox v2.0 (R11). Later releases of Matlab replace lp() with linprog() (interior-point method) that we find quite inferior to lp() on an assortment of problems; indeed, inherent limitation of numerical precision of solution to 1E-8 in linprog() causes failure in programs previously working with lp().

Given an arbitrary set of directions, this c.i. subroutine removes the conically dependent members. Yet a conically independent set returned is not necessarily unique. In that case, if desired, the set returned may be altered by reordering the set input.

% Test for c.i. of arbitrary directions in rows or columns of X.
% -Jon Dattorro

function [Xci, indep_str, how_many_depend] = conici(X,rowORcol,tol);

if nargin < 3
tol=max(size(X))*eps*norm(X);
end
if nargin < 2 | strcmp(rowORcol,'col')
rowORcol = 'col';
Xin = X;
elseif strcmp(rowORcol,'row')
Xin = X';
else
disp('Invalid rowORcol input.')
return
end
[n, N] = size(Xin);

indep_str = 'conically independent';
how_many_depend = 0;
if rank(Xin) == N
Xci = X;
return
end

count = 1;
new_N = N;
% remove zero rows or columns
for i=1:N
if chop(Xin(:,count),tol)==0
how_many_depend = how_many_depend + 1;
indep_str = 'conically Dependent';
Xin(:,count) = [ ];
new_N = new_N - 1;
else
count = count + 1;
end
end
% remove conic dependencies
count = 1;
for i=1:new_N
[a, lambda, how] = lp(zeros(newer_N-1,1),A,b,[ ],[ ],[ ],n,-1);
if ~strcmp(how,'infeasible')
how_many_depend = how_many_depend + 1;
indep_str = 'conically Dependent';
Xin(:,count) = [ ];
else
count = count + 1;
end
end
end
if strcmp(rowORcol,'col')
Xci = Xin;
else
Xci = Xin';
end


## lp()

LP     Linear programming.
X=LP(f,A,b) solves the linear programming problem:

min f'x    subject to:   Ax <= b
x

X=LP(f,A,b,VLB,VUB) defines a set of lower and upper
bounds on the design variables, X, so that the solution is
always in the range VLB <= X <= VUB.

X=LP(f,A,b,VLB,VUB,X0) sets the initial starting point to X0.

X=LP(f,A,b,VLB,VUB,X0,N) indicates that the first N constraints
defined by A and b are equality constraints.

X=LP(f,A,b,VLB,VUB,X0,N,DISPLAY) controls the level of warning
messages displayed.  Warning messages can be turned off with
DISPLAY = -1.

[X,LAMBDA]=LP(f,A,b) returns the set of Lagrangian multipliers,
LAMBDA, at the solution.

[X,LAMBDA,HOW] = LP(f,A,b) also returns a string how that
indicates error conditions at the final iteration.

LP produces warning messages when the solution is either
unbounded or infeasible.


## Map of the USA

### EDM, mapusa()

(\S\ref{USA}) \begin{verbatim} % Find map of USA using only distance information. % -Jon Dattorro % Reconstruction from EDM. clear all; close all;

load usalo; % From Matlab Mapping Toolbox \end{verbatim}\vspace{-3.7mm} {\tt\%\href{http://convexoptimization.com/TOOLS/USALO}{http://convexoptimization.com/TOOLS/USALO}} {\tt\%\href{http://www.mathworks.com/support/solutions/data/1-12MDM2.html?solution=1-12MDM2}{mathworks.com/support/solutions/data/1-12MDM2.html?solution=1-12MDM2}} \begin{verbatim} % To speed-up execution (decimate map data), make % 'factor' bigger positive integer. factor = 1; Mg = 2*factor; % Relative decimation factors Ms = factor; Mu = 2*factor;

gtlakelat = decimate(gtlakelat,Mg); gtlakelon = decimate(gtlakelon,Mg); statelat = decimate(statelat,Ms); statelon = decimate(statelon,Ms); uslat = decimate(uslat,Mu); uslon = decimate(uslon,Mu);

lat = [gtlakelat; statelat; uslat]*pi/180; lon = [gtlakelon; statelon; uslon]*pi/180; phi = pi/2 - lat; theta = lon; x = sin(phi).*cos(theta); y = sin(phi).*sin(theta); z = cos(phi);\end{verbatim}\pagebreak \begin{verbatim} % plot original data plot3(x,y,z), axis equal, axis off

lengthNaN = length(lat); id = find(isfinite(x)); X = [x(id)'; y(id)'; z(id)']; N = length(X(1,:))

% Make the distance matrix clear gtlakelat gtlakelon statelat statelon clear factor x y z phi theta conus clear uslat uslon Mg Ms Mu lat lon D = diag(X'*X)*ones(1,N) + ones(N,1)*diag(X'*X)' - 2*X'*X;

% destroy input data clear X

Vn = [-ones(1,N-1); speye(N-1)]; VDV = (-Vn'*D*Vn)/2;

clear D Vn pack

[evec evals flag] = eigs(VDV, speye(size(VDV)), 10, 'LR'); if flag, disp('convergence problem'), return, end; evals = real(diag(evals));

index = find(abs(evals) > eps*normest(VDV)*N); n = sum(evals(index) > 0); Xs = [zeros(n,1) diag(sqrt(evals(index)))*evec(:,index)'];

warning off; Xsplot=zeros(3,lengthNaN)*(0/0); warning on; Xsplot(:,id) = Xs; figure(2)

% plot map found via EDM. plot3(Xsplot(1,:), Xsplot(2,:), Xsplot(3,:)) axis equal, axis off \end{verbatim}

\subsubsection{USA map input-data decimation, {\tt decimate()}\index{decimate()}} \begin{verbatim} function xd = decimate(x,m) roll = 0; rock = 1; for i=1:length(x)

  if isnan(x(i))
roll = 0;
xd(rock) = x(i);
rock=rock+1;
else
if ~mod(roll,m)
xd(rock) = x(i);
rock=rock+1;
end
roll=roll+1;
end


end xd = xd'; \end{verbatim}

\subsection{EDM using ordinal data, {\tt omapusa()}\index{omapusa()}}\label{mouseo} (\S\ref{isomapu}) \begin{verbatim} % Find map of USA using ordinal distance information. % -Jon Dattorro clear all; close all; load usalo; % From Matlab Mapping Toolbox \end{verbatim}\vspace{-3.5mm} {\tt\%\href{http://convexoptimization.com/TOOLS/USALO}{http://convexoptimization.com/TOOLS/USALO}} {\tt\%\href{http://www.mathworks.com/support/solutions/data/1-12MDM2.html?solution=1-12MDM2}{mathworks.com/support/solutions/data/1-12MDM2.html?solution=1-12MDM2}} \begin{verbatim} factor = 1; Mg = 2*factor; % Relative decimation factors Ms = factor; Mu = 2*factor;

gtlakelat = decimate(gtlakelat,Mg); gtlakelon = decimate(gtlakelon,Mg); statelat = decimate(statelat,Ms); statelon = decimate(statelon,Ms); uslat = decimate(uslat,Mu); uslon = decimate(uslon,Mu);

lat = [gtlakelat; statelat; uslat]*pi/180; lon = [gtlakelon; statelon; uslon]*pi/180; phi = pi/2 - lat; theta = lon; x = sin(phi).*cos(theta); y = sin(phi).*sin(theta); z = cos(phi);

% plot original data plot3(x,y,z), axis equal, axis off

lengthNaN = length(lat); id = find(isfinite(x)); X = [x(id)'; y(id)'; z(id)']; N = length(X(1,:))

% Make the distance matrix clear gtlakelat gtlakelon statelat clear statelon state stateborder greatlakes clear factor x y z phi theta conus clear uslat uslon Mg Ms Mu lat lon D = diag(X'*X)*ones(1,N) + ones(N,1)*diag(X'*X)' - 2*X'*X;

% ORDINAL MDS - vectorize D count = 1; M = (N*(N-1))/2; f = zeros(M,1); for j=2:N

  for i=1:j-1
f(count) = D(i,j);
count = count + 1;
end


end % sorted is f(idx) [sorted idx] = sort(f); clear D sorted X f(idx)=((1:M).^2)/M^2;

% Create ordinal data matrix O = zeros(N,N); count = 1; for j=2:N

  for i=1:j-1
O(i,j) = f(count);
O(j,i) = f(count);
count = count+1;
end


end

clear f idx

Vn = sparse([-ones(1,N-1); eye(N-1)]); VOV = (-Vn'*O*Vn)/2;

clear O Vn pack

[evec evals flag] = eigs(VOV, speye(size(VOV)), 10, 'LR'); if flag, disp('convergence problem'), return, end; evals = real(diag(evals));

Xs = [zeros(3,1) diag(sqrt(evals(1:3)))*evec(:,1:3)'];

warning off; Xsplot=zeros(3,lengthNaN)*(0/0); warning on; Xsplot(:,id) = Xs; figure(2)

% plot map found via Ordinal MDS. plot3(Xsplot(1,:), Xsplot(2,:), Xsplot(3,:)) axis equal, axis off \end{verbatim}

\section{Rank reduction subroutine, {\tt RRf()}\index{RRf()}}\label{rrf} (\S\ref{rrpro}) \begin{verbatim} % Rank Reduction function -Jon Dattorro % Inputs are: % Xstar matrix, % affine equality constraint matrix A whose rows are in svec format. % % Tolerance scheme needs revision...

function X = RRf(Xstar,A); rand('seed',23); m = size(A,1); n = size(Xstar,1); if size(Xstar,1)~=size(Xstar,2)

  disp('Rank Reduction subroutine: Xstar not square'), pause


end toler = norm(eig(Xstar))*size(Xstar,1)*1e-9; if sum(chop(eig(Xstar),toler)<0) ~= 0

  disp('Rank Reduction subroutine: Xstar not PSD'), pause


end X = Xstar; for i=1:n

  [v,d]=signeig(X);
d(find(d<0))=0;
rho = rank(d);
for l=1:rho
R(:,l,i)=sqrt(d(l,l))*v(:,l);
end
% find Zi
svectRAR=zeros(m,rho*(rho+1)/2);
cumu=0;
for j=1:m
temp = R(:,1:rho,i)'*svectinv(A(j,:))*R(:,1:rho,i);
svectRAR(j,:) = svect(temp)';
cumu = cumu + abs(temp);
end

  % try to find sparsity pattern for Z_i
tolerance = norm(X,'fro')*size(X,1)*1e-9;
Ztem = zeros(rho,rho);
pattern = find(chop(cumu,tolerance)==0);
if isempty(pattern) % if no sparsity, do random projection
ranp = svect(2*(rand(rho,rho)-0.5));
Z(1:rho,1:rho,i)...
=svectinv((eye(rho*(rho+1)/2)-pinv(svectRAR)*svectRAR)*ranp);
else
disp('sparsity pattern found')
Ztem(pattern)=1;
Z(1:rho,1:rho,i) = Ztem;
end
phiZ = 1;
toler = norm(eig(Z(1:rho,1:rho,i)))*rho*1e-9;
if sum(chop(eig(Z(1:rho,1:rho,i)),toler)<0) ~= 0
phiZ = -1;
end
B(:,:,i) = -phiZ*R(:,1:rho,i)*Z(1:rho,1:rho,i)*R(:,1:rho,i)';
% calculate t_i^*
t(i) = max(phiZ*eig(Z(1:rho,1:rho,i)))^-1;
tolerance = norm(X,'fro')*size(X,1)*1e-6;
if chop(Z(1:rho,1:rho,i),tolerance)==zeros(rho,rho)
break
else
X = X + t(i)*B(:,:,i);
end


end \end{verbatim}

\pagebreak \subsection{{\tt svect()}\index{svect()}} \begin{verbatim} % Map from symmetric matrix to vector % -Jon Dattorro

function y = svect(Y,N)

if nargin == 1

  N=size(Y,1);


end

y = zeros(N*(N+1)/2,1); count = 1; for j=1:N

  for i=1:j
if i~=j
y(count) = sqrt(2)*Y(i,j);
else
y(count) = Y(i,j);
end
count = count + 1;
end


end \end{verbatim}

\newpage \subsection{{\tt svectinv()}\index{svectinv()}} \begin{verbatim} % convert vector into symmetric matrix. m is dim of matrix. % -Jon Dattorro function A = svectinv(y)

m = round((sqrt(8*length(y)+1)-1)/2); if length(y) ~= m*(m+1)/2

  disp('dimension error in svectinv()');
pause


end

A = zeros(m,m); count = 1; for j=1:m

  for i=1:m
if i<=j
if i==j
A(i,i) = y(count);
else
A(i,j) = y(count)/sqrt(2);
A(j,i) = A(i,j);
end
count = count+1;
end
end


end \end{verbatim}

\newpage \section{Sturm's procedure\index{procedure!Sturm}\index{procedure!dyad-decomposition}}\label{spu} This is a demonstration program that can easily be transformed to a subroutine for decomposing positive semidefinite matrix $X\,$.\, This procedure provides a nonorthogonal alternative (\S\ref{spu2}) to eigen decomposition. That particular decomposition obtained is dependent on choice of matrix $A\,$.

\begin{verbatim} % Sturm procedure to find dyad-decomposition of X -Jon Dattorro clear all

N = 4; r = 2; X = 2*(rand(r,N)-0.5); X = X'*X;

t = null(svect(X)'); A = svectinv(t(:,1));

% Suppose given matrix A is positive semidefinite %[v,d] = signeig(X); %d(1,1)=0; d(2,2)=0; d(3,3)=pi; %A = v*d*v';

tol = 1e-8; Y = X; y = zeros(size(X,1),r); rho = r; for k=1:r

   [v,d] = signeig(Y);
v = v*sqrt(chop(d,1e-14));
viol = 0;
j = [ ];
for i=2:rho
if chop((v(:,1)'*A*v(:,1))*(v(:,i)'*A*v(:,i)),tol) ~= 0
viol = 1;
end
if (v(:,1)'*A*v(:,1))*(v(:,i)'*A*v(:,i)) < 0
j = i;
break
end
end
if ~viol
y(:,k) = v(:,1);
else
if isempty(j)
disp('Sturm procedure taking default j'), j = 2; return
end  % debug
alpha = (-2*(v(:,1)'*A*v(:,j)) + sqrt((2*v(:,1)'*A*v(:,j)).^2 ...
-4*(v(:,j)'*A*v(:,j))*(v(:,1)'*A*v(:,1))))/(2*(v(:,j)'*A*v(:,j)));
y(:,k) = (v(:,1) + alpha*v(:,j))/sqrt(1+alpha^2);
if chop(y(:,k)'*A*y(:,k),tol) ~= 0
alpha = (-2*(v(:,1)'*A*v(:,j)) - sqrt((2*v(:,1)'*A*v(:,j)).^2 ...
-4*(v(:,j)'*A*v(:,j))*(v(:,1)'*A*v(:,1))))/(2*(v(:,j)'*A*v(:,j)));
y(:,k) = (v(:,1) + alpha*v(:,j))/sqrt(1+alpha^2);
if chop(y(:,k)'*A*y(:,k),tol) ~= 0
disp('Zero problem in Sturm!'), return
end  % debug
end
end
Y = Y - y(:,k)*y(:,k)';
rho = rho - 1;


end z = zeros(size(y)); e = zeros(N,N); for i=1:r

   z(:,i) = y(:,i)/norm(y(:,i));
e(i,i) = norm(y(:,i))^2;


end lam = diag(e); [junk id] = sort(real(lam)); id = id(length(id):-1:1); z = [z(:,id(1:r)) null(z')]  % Sturm e = diag(lam(id)) [v,d] = signeig(X)  % eigenvalue decomposition X-z*e*z' traceAX = trace(A*X) \end{verbatim}

\newpage \section{Convex Iteration demonstration}\label{berg} We demonstrate implementation of a rank constraint in a semidefinite Boolean feasibility problem from \S\ref{Booleanlt}. It requires {\tt CVX}, \cite{cvx} an intuitive {\sc Matlab} interface for interior-point method solvers.

There are a finite number \,\mbox{$2^{N=_{}50}\!\approx\!{\tt 1E15}$}\, of binary \hbox{vectors \,$x\,$}.\, The feasible set of semidefinite program (\ref{ll041}) is the intersection of an elliptope with \,\mbox{$M\!=\!10$}\, halfspaces in vectorized composite $_{}G\,$.\, Size of the optimal rank$_{}$-$1$ solution set is proportional to the positive factor scaling vector \,{\tt b\,}.\, The smaller that optimal Boolean solution set, the harder this problem is to solve; indeed, it can be made as small as one point. That scale factor and initial state of random number generators, making matrix \,{\tt A}\, and \hbox{vector \,{\tt b\,}},\, are selected to demonstrate Boolean solution to one instance in a few iterations \textbf{(}a few seconds\textbf{)}, whereas sequential binary search takes one hour to test \hbox{$25.7$ million} vectors before finding one Boolean solution feasible to nonconvex \hbox{problem (\ref{obp})}. (Other parameters can be selected to realize a reversal of these timings.)

\begin{verbatim} % Discrete optimization problem demo. % -Jon Dattorro, June 4, 2007 % Find x\in{-1,1}^N such that Ax <= b clear all; format short g; M = 10; N = 50; randn('state',0); rand('state',0); A = randn(M,N); b = rand(M,1)*5;

disp('Find binary solution by convex iteration:') tic Y = zeros(N+1); count = 1; traceGY = 1e15; cvx_precision([1e-12, 1e-4]); cvx_quiet(true);\end{verbatim}\pagebreak \begin{verbatim} while 1

   cvx_begin  % requires CVX Boyd
variable X(N,N) symmetric;
variable x(N,1);
G = [X,  x;
x', 1];
minimize(trace(G*Y));
diag(X) == 1;
G == semidefinite(N+1);
A*x <= b;
cvx_end

   [v,d,q] = svd(G);
Y = v(:,2:N+1)*v(:,2:N+1)';
rankG = sum(diag(d) > max(diag(d))*1e-8)
oldtrace = traceGY;
traceGY = trace(G*Y)
if rankG == 1
break
end

   if round((oldtrace - traceGY)*1e3) == 0
disp('STALLED');disp(' ');
Y = -v(:,2:N+1)*(v(:,2:N+1)' + randn(N,1)*v(:,1)');
end
count = count + 1;


end x count toc disp('Ax <= b , x\in{-1,1}^N')\end{verbatim}\pagebreak \begin{verbatim} disp(' ');disp('Combinatorial search for a feasible binary solution:') tic for i=1:2^N

   binary = str2num(dec2bin(i-1)');
binary(find(~binary)) = -1;
y = [-ones(N-length(binary),1); binary];
if sum(A*y <= b) == M
disp('Feasible binary solution found.')
y
break
end


end toc \end{verbatim}

\newpage \sectionTemplate:\sc fast max cut\label{fmck} We use the graph generator (C program) {\sc rudy} written by Giovanni Rinaldi \cite{Rinaldi} which can be found at \,\href{http://convexoptimization.com/TOOLS/RUDY}{\tt http://convexoptimization.com/TOOLS/RUDY}\, together with graph data. (\S\ref{fastmc}) \begin{verbatim} % fast max cut, Jon Dattorro, July 2007, http://convexoptimization.com clear all; format short g; tic fid = fopen('graphs12','r'); average = 0; NN = 0; s = fgets(fid); cvx_precision([1e-12, 1e-4]); cvx_quiet(true); w = 1000; while s ~= -1

   s = str2num(s);
N = s(1);
A = zeros(N);
for i=1:s(2)
s = str2num(fgets(fid));
A(s(1),s(2)) = s(3);
A(s(2),s(1)) = s(3);
end
Q = (diag(A*ones(N,1)) - A)/4;
W = zeros(N);
traceXW = 1e15;
while 1
cvx_begin                     % CVX Boyd
variable X(N,N) symmetric;
maximize(trace(Q*X) - w*trace(W*X));
X == semidefinite(N);
diag(X) == 1;
cvx_end
[v,d,q] = svd(X);
W = v(:,2:N)*v(:,2:N)';
rankX = sum(diag(d) > max(diag(d))*1e-8)
oldtrace = traceXW;
traceXW = trace(X*W)
if (rankX == 1)
break
end
if round((oldtrace - traceXW)*1e3) <= 0
disp('STALLED');disp(' ')
W = -v(:,2:N)*(v(:,2:N)' + randn(N-1,1)*v(:,1)');
end
end
x = sqrt(d(1,1))*v(:,1)
disp(' ');
disp('Combinatorial search for optimal binary solution...')
maxim = -1e15;
ymax = zeros(N,1);
for i=1:2^N
binary = str2num(dec2bin(i-1)');
binary(find(~binary)) = -1;
y = [-ones(N-length(binary),1); binary];
if y'*Q*y > maxim
maxim = y'*Q*y;
ymax = y;
end
end
if (maxim == 0) && (abs(trace(Q*X)) <= 1e-8)
optimality_ratio = 1
elseif maxim <= 0
optimality_ratio = maxim/trace(Q*X)
else
optimality_ratio = trace(Q*X)/maxim
end
ymax
average = average + optimality_ratio;
NN = NN + 1
running_average = average/NN
toc, disp(' ')
s = fgets(fid);


end \end{verbatim}

\section{Signal dropout problem}\label{sdpr} (\S\ref{ssdp}) Requires {\tt CVX}. \cite{cvx} \begin{verbatim} %signal dropout problem -Jon Dattorro clear all close all N = 500; k = 4;  % cardinality Phi = dctmtx(N)';  % DCT basis

successes = 0; total = 0; logNormOfDifference_rate = 0; SNR_rate = 0; for i=1:1

   close all
SNR = 0;
na = .02;
SNR10 = 10;
while SNR <= SNR10
xs = zeros(N,1);        % initialize x*
p = randperm(N);        % calls rand
xs(p(1:k)) = 10^(-SNR10/20) + (1-10^(-SNR10/20))*rand(k,1);
s = Phi*xs;
noise = na*randn(size(s));
ll = 130;
ul = N-ll+1;
SNR = 20*log10(norm([s(1:ll); s(ul:N)])/norm(noise));
end

   normof = 1e15;
count = 1;
y = zeros(N,1);
cvx_quiet(true);\end{verbatim}\pagebreak


\begin{verbatim}

   while 1
cvx_begin
variable x(N);
f = Phi*x;
minimize(x'*y + norm([f(1:ll)-(s(1:ll)+noise(1:ll));
f(ul:N)-(s(ul:N)+noise(ul:N))]));
x >= 0;
cvx_end

       if ~(strcmp(cvx_status,'Solved') ||...
strcmp(cvx_status,'Inaccurate/Solved'))
disp('cit failed')
return
end

       cvx_begin
variable y(N);
minimize(x'*y);
0 <= y; y <= 1;
y'*ones(N,1) == N-k;
cvx_end

       if ~(strcmp(cvx_status,'Solved') ||...
strcmp(cvx_status,'Inaccurate/Solved'))
disp('Fan failed')
return
end

       cardx = sum(x > max(x)*1e-8)
traceXW = x'*y
oldnorm = normof;
normof = norm([f(1:ll)-(s(1:ll)+noise(1:ll));
f(ul:N)-(s(ul:N)+noise(ul:N))]);
if (cardx <= k) && (abs(oldnorm - normof) <= 1e-8)
break
end
count = count + 1;
end
count
figure(1);plot([s(1:ll)+noise(1:ll); noise(ll+1:ul-1);
s(ul:N)+noise(ul:N)]);
hold on;  plot([s(1:ll)+noise(1:ll); zeros(ul-ll-1,1);
s(ul:N)+noise(ul:N)],'k')
V = axis;
figure(2);plot(f,'r');
hold on; plot([s(1:ll)+noise(1:ll); noise(ll+1:ul-1);
s(ul:N)+noise(ul:N)]);
axis(V);
logNormOfDifference = 20*log10(norm(f-s)/norm(s))
SNR
figure(3);plot(f,'r'); hold on; plot(s);axis(V);
figure(4);plot(f-s);axis(V);
temp = sort([find(chop(x,max(x)*1e-8)); zeros(k-cardx,1)])...
- sort(p(1:k))'
if ~sum(temp)
successes = successes + 1;
end
total = total + 1
success_avg = successes/total
logNormOfDifference_rate = logNormOfDifference_rate...
+ logNormOfDifference;
SNR_rate = SNR_rate + SNR;
logNormOfDifferenceAvg = logNormOfDifference_rate/total
SNR_avg = SNR_rate/total, disp(' ')


end successes </pre>