# Matlab for Convex Optimization & Euclidean Distance Geometry

(Difference between revisions)
 Revision as of 00:13, 26 February 2009 (edit)← Previous diff Revision as of 19:05, 5 March 2009 (edit) (undo) (→Dx())Next diff → Line 197: Line 197: one = ones(N,1); one = ones(N,1); - del = diag(X'*X); + XTX = X'*X; - D = del*one' + one*del' - 2*X'*X; + del = diag(XTX); + D = del*one' + one*del' - 2*XTX; - == conic independence == == conic independence ==

## Revision as of 19:05, 5 March 2009

These MATLAB programs come from the book Convex Optimization & Euclidean Distance Geometry, by Dattorro, which is available for download online.

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);

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


## 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.

### conici()

% Remove c.i. directions in rows or columns of X.  -Jon Dattorro
% function [Xci, indep_str, how_many_depend, kept] = conici(X,'rowORcol',tolerance);

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

if nargin < 3
tol=max(size(X))*eps*normest(X);
end
if nargin < 2 || strcmp(rowORcol,'col') || isempty(rowORcol)
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;
kept = [1:N]';
if rank(Xin,tol) == N
Xci = chop(X,tol);
return
end

count = 1;
new_N = N;
%remove zero columns
for i=1:new_N
if chop(Xin(:,count),tol)==0
how_many_depend = how_many_depend + 1;
indep_str = 'conically Dependent';
Xin(:,count) = [ ];
kept(count) = [ ];
new_N = new_N - 1;
else
count = count + 1;
end
end
%remove identical columns
D = sqrt(Dx(Xin));
T = tril(ones(new_N,new_N))*normest(D);
ir = find(D+T < tol);
if ~isempty(ir)
[iir jir] = ind2sub(size(D),ir);
indep_str = 'conically Dependent';
sizebefore = size(Xin);
Xin(:,jir) = [ ];
sizeafter = size(Xin);
kept(jir) = [ ];
new_N = size(Xin,2);
how_many_depend = how_many_depend + sizebefore(2)-sizeafter(2);
end
%remove conic dependencies
count = 1;
[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) = [ ];
kept(count) = [ ];
else
count = count + 1;
end
end
end
if strcmp(rowORcol,'col')
Xci = chop(Xin, tol);
else
Xci = chop(Xin',tol);
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()

% Find map of USA using only distance information.
% -Jon Dattorro
% Reconstruction from EDM.
clear all;
close all;

load usalo; % From Matlab Mapping Toolbox

% 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);

% 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


#### USA map input-data decimation, decimate()

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';


### EDM using ordinal data, omapusa()

% Find map of USA using ordinal distance information.
% -Jon Dattorro
clear all;
close all;

load usalo; % From Matlab Mapping Toolbox

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


## Rank reduction subroutine, RRf()

% 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


### svect()

% 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


### svectinv()

% 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


## Sturm & Zhang's procedure for constructing dyad-decomposition

This is a demonstration program that can easily be transformed to a subroutine for decomposing positive semidefinite matrix $LaTeX: \,X\,$.

This procedure provides a nonorthogonal alternative to eigen decomposition.

That particular decomposition obtained is dependent on choice of matrix $LaTeX: \,A\,$.

% 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)


## Convex Iteration demonstration - Boolean feasibility

We demonstrate implementation of a rank constraint in a semidefinite Boolean feasibility problem.

It requires CVX, an intuitive Matlab interface for interior-point method solvers.

There are a finite number $LaTeX: \,2^{N=_{}50}\!\approx\!1E15\,$ of binary vectors $LaTeX: \,x\,$.

The feasible set of the semidefinite program from the book is the intersection of an elliptope with $LaTeX: \,M\!=\!10\,$ halfspaces in vectorized composite $LaTeX: \,G\,$.

Size of the optimal rank-1 solution set is proportional to the positive factor scaling vector 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 A and vector b, are selected to demonstrate Boolean solution to one instance in a few iterations (a few seconds);

whereas sequential binary search takes one hour to test 25.7 million vectors before finding one Boolean solution feasible to the nonconvex problem from the book.

(Other parameters can be selected to realize a reversal of these timings.)

% 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);

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')

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


## Convex Iteration rank-1

This program demonstrates how any semidefinite problem with a rank-k constraint is equivalently transformed into a problem sequence having a rank-1 constraint; discussed at the end of Chapter 4.

%Convex Iteration Rank-1,  -Jon Dattorro October 2007
% find        X
% subject to  A vec X = b
%             X >= 0
%             rank X <= k

% X\in\reals^{N\times N}
clear all
N=26;
M=10;
randn('seed',100);
A = randn(M,N*N);
b = randn(M,1);

r = 2;
W = ones(r*N);
Q11 = zeros(N,N);
Q22 = zeros(N,N);
count = 1;
traceGW = 1e15;
normof = 1e15;
w1 = 5;
cvx_quiet(true);
cvx_precision([1e-10 1e-4]);
while 1
cvx_begin           %Requires CVX Boyd
variable L(r,1);
X = L(1)*Q11 + L(2)*Q22;
minimize(norm(A*X(:) - b));
L >= 0;
cvx_end

% find Q
cvx_begin
variable Q11(N,N) symmetric;
variable Q12(N,N);
variable Q22(N,N) symmetric;

trace(Q11) == 1;
trace(Q22) == 1;
trace(Q12) == 0;

G = [Q11,  Q12;
Q12', Q22];
G == semidefinite(r*N);

X = L(1)*Q11 + L(2)*Q22;

minimize(w1*trace(G*W) + norm(A*X(:) - b));
cvx_end

[v,d,q] = svd(full(G));
W = v(:,2:r*N)*v(:,2:r*N)';
rankG = sum(diag(d) > max(diag(d))*1e-6)
oldtraceGW = traceGW;
traceGW = trace(G*W)
oldnorm = normof;
normof = norm(A*X(:) - b)
if (rankG == 1) && (norm(A*X(:) - b) < 1e-6)
break
end

if round((oldtraceGW - traceGW)*1e3) <= 0 && round((oldnorm - normof)*1e3) <= 0
disp('STALLED');disp(' ')
W = -v(:,2:r*N)*(v(:,2:r*N)' + randn(r*N-1,1)*v(:,1)');
end
count = count + 1;
end
count


## fast max cut

We use the graph generator (C program) rudy written by Giovanni Rinaldi which can be found at http://convexoptimization.com/TOOLS/RUDY together with graph data.

% 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


## Signal dropout problem

Requires CVX (Boyd).

%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);

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


## Compressive Sampling of Images by Convex Iteration $LaTeX: -$ Shepp-Logan phantom

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Center for Advanced Imaging Research
% Mayo Clinic
% 4/7/2007
%   Rewritten for cardinality minimization by Convex Iteration.
%   -Jon Dattorro, July 2008.
%   Example. "Compressive sampling of a phantom" from Convex Optimization & Euclidean Distance Geometry.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
%%% Generate Shepp-Logan phantom %%%
N = 256;
image = chop(phantom(N));
if  mod(N,2), disp('N must be even for symmMap()'), return, end

P = 10;                         %noninteger bends lines at origin.
PHI = fftshift(symmMap(N,P));   %left justification of DC-centric radial sampling pattern

f = vect(real(ifft2(PHI.*fft2(image))));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
lambda    = 1e8; 	% fidelity parameter on equality
tol_inner = 1e-2; 	% inner loop exit tolerance
tol_outer = 1e-5; 	% inner loop exit tolerance
maxCGiter = 35;   	% max Conjugate Gradient iterations

%%% Build differential matrices %%%
Psi1 = kron(speye(N) - spdiags(ones(N,1),-1,N,N), speye(N))';
Psi2 = kron(speye(N), speye(N) - spdiags(ones(N,1),-1,N,N));
Psi = [Psi1; Psi1'; Psi2; Psi2'];

close all
figure, colormap(gray), axis image, axis off, set(gca,'position',[0 0 1 1]); set(gcf,'position',[600 1300 512 512])

count = 0;
u = vectinv(f);
old_u = 0;
last_u = 0;
y = ones(4*N^2,1);
card = 5092;        % in vicinity of minimum cardinality which is 5092 at N=256; e.g., 8000 works
tic
while 1
while 1
count = count + 1;
%%%%% r0 %%%%%
tmp = Psi'*spdiags(y./(abs(Psi*u(:))+1e-3),0,4*N^2,4*N^2)*Psi;
r = tmp*u(:) + lambda*vect(real(ifft2(PHI.*fft2(u - vectinv(f)))));

%%%%% Inversion via Conjugate Gradient %%%%%
p = 0;
beta = 0;
for i=1:maxCGiter
p = beta*p - r;
Gp = tmp*p + lambda*vect(real(ifft2(PHI.*fft2(vectinv(p)))));
rold = r'*r;
alpha = rold/(p'*Gp);
if norm(alpha*p)/norm(u(:)) < 1e-13
break
end
u(:) = u(:) + alpha*p;
r = r + alpha*Gp;
beta = (r'*r)/rold;
end
if norm(u(:)) > 10*norm(f)
disp('CG exit')
disp(strcat('norm(u(:))=',num2str(norm(u(:)))))
u = old_u;
end
%%%%% display intermediate image %%%%%
imshow(real(u),[])
pause(0.04)
disp(count)

%%% test for fixed point %%%
if norm(u(:) - old_u(:))/norm(u(:)) < tol_inner
break
end
old_u = u;
end
%%% measure cardinality %%%
err = 20*log10(norm(image-u,'fro')/norm(image,'fro'));
disp(strcat('error=',num2str(err),' dB'))
%%% new direction vector %%%
[x_sorted, indices] = sort(abs(Psi*u(:)),'descend');
cardest = 4*N^2 - length(find(x_sorted < 1e-3*max(x_sorted)));
disp(strcat('cardinality=',num2str(cardest)))
y = ones(4*N^2,1);
y(indices(1:card)) = 0;

if norm(u(:) - last_u(:))/norm(u(:)) < tol_outer
break
end
last_u = u;
end
toc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% results %%%
figure
imshow(image,[])
colormap(gray)
axis image
axis off

figure
imshow(fftshift(PHI),[])
colormap(gray)
axis image
axis off

figure
imshow(abs(vectinv(f)),[])
colormap(gray)
axis image
axis off

figure
imshow(real(u),[])
colormap(gray)
axis image
axis off

disp(strcat('ratio_Fourier_samples=',num2str(sum(sum(PHI))/N^2)))


### symmMap()

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Center for Advanced Imaging Research
% Mayo Clinic
% 4/7/2007
%   Modified for horizontal and vertical symmetry.
%   -Jon Dattorro July 2008
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function SM2 = symmMap(N0,P)
N = ceil(N0*sqrt(2)/2)*2;
sampleMap = zeros(N,N);
slice     = zeros(N/2+1,2);
rot_slice = zeros(N/2+1,2);
dTheta = pi/P;
for l=1:N/2+1
x = N/2+1;
y = l;
slice(l,1) = x;
slice(l,2) = y;
end
for p = 0:2*P
theta = p*dTheta;
for n = 1:N/2
x = slice(n,1)-N/2-1;
y = slice(n,2)-N/2-1;
x2 = round( x*cos(theta)+y*sin(theta)+N/2+1);
y2 = round(-x*sin(theta)+y*cos(theta)+N/2+1);
sampleMap(y2,x2) = 1;
end
end
SM = sampleMap(1:N,1:N);
SM(N/2+1,N/2+1) = 1;

Nc  = N/2;
N0c = N0/2;

SM2 = SM(Nc-N0c+1:Nc+N0c, Nc-N0c+1:Nc+N0c);

% make vertically and horizontally symmetric
[N1,N1] = size(SM2);
Xi = fliplr(eye(N1-1));
SM2 = round((SM2 + [   SM2(1,1)        SM2(1,2:N1)*Xi;
Xi*SM2(2:N1,1)  Xi*SM2(2:N1,2:N1)*Xi])/2);


### vect()

function y = vect(A);
y = A(:);


### vectinv()

%convert vector into matrix.  m is dim of matrix.
function A = vectinv(y)

m = round(sqrt(length(y)));
if length(y) ~= m^2
disp('dimension error in vectinv()');
return
end

A = reshape(y,m,m);