file is outputcvxsept22.txt
disp('Sept 22/09 ')
Sept 22/09 
startupcvx   % personal startup command to add correct paths
  addpath c:\software.d\cvx\builtins
    addpath c:\software.d\cvx\commands
    addpath c:\software.d\cvx\functions
    addpath c:\software.d\cvx\lib
    addpath c:\software.d\cvx\structures
    addpath c:\software.d\cvx\examples
    addpath c:\software.d\cvx
disp('try out the quickstart command')
try out the quickstart command
echo off

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.1: LEAST SQUARES %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Input data
m = 16; n = 8;
A = randn(m,n);
b = randn(m,1);

% Matlab version
x_ls = A \ b;

% cvx version
cvx_begin
    variable x(n);
    minimize( norm(A*x-b) );
cvx_end

echo off

Results:
--------
norm(A*x_ls-b): 2.0354
norm(A*x-b):    2.0354
cvx_optval:     2.0354
cvx_status:     Solved

Verify that x_ls == x:
   x_ls  = [ -0.2628  0.8828 -0.0734 -1.0844  0.3249 -0.3330  0.0603  0.3802 ]
   x     = [ -0.2628  0.8828 -0.0734 -1.0844  0.3249 -0.3330  0.0603  0.3802 ]
Residual vector:
   A*x-b = [ -0.3262 -0.0070 -0.9543  0.2447 -0.6418 -0.3426 -0.1870  0.2960  0.6024 -0.0440  0.6238 -0.7399  0.0849  0.9323  0.4799 -0.0762 ]
 
Press Enter/Return for the next example...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.2: BOUND-CONSTRAINED LEAST SQUARES %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% More input data
bnds = randn(n,2);
l = min( bnds, [] ,2 );
u = max( bnds, [], 2 );

if has_quadprog,
    % Quadprog version
    x_qp = quadprog( 2*A'*A, -2*A'*b, [], [], [], [], l, u );
Optimization terminated: relative function value changing by
 less than OPTIONS.TolFun.

% cvx version
cvx_begin
    variable x(n);
    minimize( norm(A*x-b) );
    subject to
        x >= l;
        x <= u;
cvx_end

echo off

Results:
--------
norm(A*x_qp-b): 4.1334
norm(A*x-b):    4.1334
cvx_optval:     4.1334
cvx_status:     Solved

Verify that l <= x_qp == x <= u:
   l     = [ -0.5618  0.2760 -0.2277 -0.0290 -0.9287  0.4520  0.1014 -0.3658 ]
   x_qp  = [ -0.0910  0.2918  0.2746 -0.0290  0.0828  0.4520  0.1014  0.6919 ]
   x     = [ -0.0910  0.2918  0.2746 -0.0290  0.0828  0.4520  0.1014  0.6919 ]
   u     = [ -0.0910  0.7395  0.9403  0.1842  0.0828  0.7450  2.4881  0.6919 ]
Residual vector:
   A*x-b = [ -0.1209  0.2155 -1.0903 -0.1312 -2.0952  1.6798  0.3784 -0.5592  1.0411  0.6937  1.6036 -0.0045  0.9935  0.2156  1.2186 -1.2228 ]
 
Press Enter/Return for the next example...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.3: OTHER NORMS AND FUNCTIONS: INFINITY NORM %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if has_linprog,
    % linprog version
    f    = [ zeros(n,1);  1          ];
    Ane  = [ +A,          -ones(m,1) ; ...
             -A,          -ones(m,1) ];
    bne  = [ +b;          -b         ];
    xt   = linprog(f,Ane,bne);
Optimization terminated.
    x_lp = xt(1:n,:);

% cvx version
cvx_begin
    variable x(n);
    minimize( norm(A*x-b,Inf) );
cvx_end

echo off

Results:
--------
norm(A*x_lp-b,Inf): 0.7079
norm(A*x-b,Inf):    0.7079
cvx_optval:         0.7079
cvx_status:         Solved

Verify that x_lp == x:
   x_lp  = [ -0.0944  0.8498 -0.1119 -1.1311  0.3804 -0.3017  0.2201  0.2488 ]
   x     = [ -0.0944  0.8498 -0.1119 -1.1311  0.3804 -0.3017  0.2201  0.2488 ]
Residual vector; verify that the peaks match the objective (0.7079):
   A*x-b = [ -0.0431 -0.0539 -0.7079  0.7079 -0.7079 -0.7079 -0.1800  0.5049  0.7079 -0.0040  0.7079 -0.7079 -0.1010  0.7079  0.7079 -0.2187 ]
 
Press Enter/Return for the next example...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.3: OTHER NORMS AND FUNCTIONS: ONE NORM %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if has_linprog,
    % Matlab version
    f    = [ zeros(n,1); ones(m,1);  ones(m,1)  ];
    Aeq  = [ A,          -eye(m),    +eye(m)    ];
    lb   = [ -Inf*ones(n,1);  zeros(m,1); zeros(m,1) ];
    xzz  = linprog(f,[],[], Aeq,b,lb,[]);
Optimization terminated.
    x_lp = xzz(1:n,:);

% cvx version
cvx_begin
    variable x(n);
    minimize( norm(A*x-b,1) );
cvx_end

echo off

Results:
--------
norm(A*x_lp-b,1): 5.3359
norm(A*x-b,1): 5.3359
cvx_optval: 5.3359
cvx_status: Solved

Verify that x_lp == x:
   x_lp  = [ -0.3550  0.8934 -0.0375 -1.1827  0.1694 -0.3870 -0.2148  0.6712 ]
   x     = [ -0.3550  0.8934 -0.0375 -1.1827  0.1694 -0.3870 -0.2148  0.6712 ]
Residual vector; verify the presence of several zero residuals:
   A*x-b = [ -0.7666  0.0129 -1.4977  0.0000 -0.5074 -0.0000 -0.0000  0.0357  0.0000  0.0000  0.0299 -1.0842 -0.0000  1.4013  0.0000 -0.0000 ]
 
Press Enter/Return for the next example...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.3: OTHER NORMS AND FUNCTIONS: LARGEST-K NORM %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% cvx specification
k = 5;
cvx_begin
    variable x(n);
    minimize( norm_largest(A*x-b,k) );
cvx_end

echo off

Results:
--------
norm_largest(A*x-b,k): 3.5394
cvx_optval: 3.5394
cvx_status: Solved

Optimal vector:
   x     = [ -0.0944  0.8498 -0.1119 -1.1311  0.3804 -0.3017  0.2201  0.2488 ]
Residual vector; verify a tie for 5-th place ( 0.7079):
   A*x-b = [ -0.0431 -0.0539 -0.7079  0.7079 -0.7079 -0.7079 -0.1800  0.5049  0.7079 -0.0040  0.7079 -0.7079 -0.1010  0.7079  0.7079 -0.2187 ]
 
Press Enter/Return for the next example...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.3: OTHER NORMS AND FUNCTIONS: HUBER PENALTY %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% cvx specification
cvx_begin
    variable x(n);
    minimize( sum(huber(A*x-b)) );
cvx_end

echo off

Results:
--------
sum(huber(A*x-b)): 4.1428
cvx_optval: 4.1428
cvx_status: Solved

Optimal vector:
   x     = [ -0.2628  0.8828 -0.0734 -1.0844  0.3249 -0.3330  0.0603  0.3802 ]
Residual vector:
   A*x-b = [ -0.3262 -0.0070 -0.9543  0.2447 -0.6418 -0.3426 -0.1870  0.2960  0.6024 -0.0440  0.6238 -0.7399  0.0849  0.9323  0.4799 -0.0762 ]
 
Press Enter/Return for the next example...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.4: OTHER CONSTRAINTS %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% More input data
p = 4;
C = randn(p,n);
d = randn(p,1);

% cvx specification
cvx_begin
    variable x(n);
    minimize( norm(A*x-b) );
    subject to
        C*x == d;
        norm(x,Inf) <= 1;
cvx_end

echo off

Results:
--------
norm(A*x-b): 5.9545
cvx_optval: 5.9545
cvx_status: Solved

Optimal vector:
   x     = [  0.1173  1.0000  0.0979 -0.1256  0.5255  0.7102 -0.0127  0.8397 ]
Residual vector:
   A*x-b = [ -0.7438  0.0763 -2.0376 -1.3110 -2.3956  0.2175 -1.2595 -0.7583  1.6970  1.9857  0.8180  1.0527  2.5150 -1.2068  2.1631 -0.1360 ]
Equality constraints:
   C*x   = [  1.0035 -2.6761  0.0168 -1.4432 ]
   d     = [  1.0035 -2.6761  0.0168 -1.4432 ]
Press Enter/Return for the next example...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.5: AN OPTIMAL TRADEOFF CURVE %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The basic problem:
% cvx_begin
%     variable x(n);
%     minimize( norm(A*x-b)+gamma(k)*norm(x,1) );
% cvx_end
echo off
 
Generating tradeoff curve...
   gamma       norm(x,1)    norm(A*x-b)
---------------------------------------
1.0000e-002   3.3796e+000   2.0355e+000
1.6238e-002   3.3658e+000   2.0357e+000
2.6367e-002   3.3433e+000   2.0362e+000
4.2813e-002   3.3068e+000   2.0374e+000
6.9519e-002   3.2473e+000   2.0408e+000
1.1288e-001   3.1498e+000   2.0497e+000
1.8330e-001   2.9878e+000   2.0737e+000
2.9764e-001   2.7617e+000   2.1280e+000
4.8329e-001   2.3445e+000   2.2924e+000
7.8476e-001   1.4081e+000   2.8895e+000
1.2743e+000   3.7687e-001   3.9510e+000
2.0691e+000   1.0114e-008   4.4813e+000
3.3598e+000   5.3229e-009   4.4813e+000
5.4556e+000   1.9095e-009   4.4813e+000
8.8587e+000   5.4562e-010   4.4813e+000
1.4384e+001   2.6802e-010   4.4813e+000
2.3357e+001   3.7101e-010   4.4813e+000
3.7927e+001   1.0869e-010   4.4813e+000
6.1585e+001   2.5093e-011   4.4813e+000
1.0000e+002   2.8799e-011   4.4813e+000
Done. (Check out the graph!)