has_quadprog = exist( 'quadprog' );
has_quadprog = has_quadprog == 2 | has_quadprog == 3;
has_linprog  = exist( 'linprog' );
has_linprog  = has_linprog == 2 | has_linprog == 3;
rnstate = randn( 'state' ); randn( 'state', 1 );
s_quiet = cvx_quiet(true);
s_pause = cvx_pause(false);
cvx_clear; echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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

% Compare
disp( sprintf( '\nResults:\n--------\nnorm(A*x_ls-b): %6.4f\nnorm(A*x-b):    %6.4f\ncvx_optval:     %6.4f\ncvx_status:     %s\n', norm(A*x_ls-b), norm(A*x-b), cvx_optval, cvx_status ) );
disp( 'Verify that x_ls == x:' );
disp( [ '   x_ls  = [ ', sprintf( '%7.4f ', x_ls ), ']' ] );
disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
disp( 'Residual vector:' );
disp( [ '   A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( ' ' );

try, input( 'Press Enter/Return for the next example...' ); clc; catch, end
echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 );
else,
    % quadprog not present on this system.
end

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

echo off

% Compare
if has_quadprog,
    disp( sprintf( '\nResults:\n--------\nnorm(A*x_qp-b): %6.4f\nnorm(A*x-b):    %6.4f\ncvx_optval:     %6.4f\ncvx_status:     %s\n', norm(A*x_qp-b), norm(A*x-b), cvx_optval, cvx_status ) );
    disp( 'Verify that l <= x_qp == x <= u:' );
    disp( [ '   l     = [ ', sprintf( '%7.4f ', l ), ']' ] );
    disp( [ '   x_qp  = [ ', sprintf( '%7.4f ', x_qp ), ']' ] );
    disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
    disp( [ '   u     = [ ', sprintf( '%7.4f ', u ), ']' ] );
else,
    disp( sprintf( '\nResults:\n--------\nnorm(A*x-b): %6.4f\ncvx_optval:  %6.4f\ncvx_status:  %s\n', norm(A*x-b), cvx_optval, cvx_status ) );
    disp( 'Verify that l <= x <= u:' );
    disp( [ '   l     = [ ', sprintf( '%7.4f ', l ), ']' ] );
    disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
    disp( [ '   u     = [ ', sprintf( '%7.4f ', u ), ']' ] );
end
disp( 'Residual vector:' );
disp( [ '   A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( ' ' );

try, input( 'Press Enter/Return for the next example...' ); clc; catch, end
echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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);
    x_lp = xt(1:n,:);
else,
    % linprog not present on this system.
end

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

echo off

% Compare
if has_linprog,
    disp( sprintf( '\nResults:\n--------\nnorm(A*x_lp-b,Inf): %6.4f\nnorm(A*x-b,Inf):    %6.4f\ncvx_optval:         %6.4f\ncvx_status:         %s\n', norm(A*x_lp-b,Inf), norm(A*x-b,Inf), cvx_optval, cvx_status ) );
    disp( 'Verify that x_lp == x:' );
    disp( [ '   x_lp  = [ ', sprintf( '%7.4f ', x_lp ), ']' ] );
    disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
else,
    disp( sprintf( '\nResults:\n--------\nnorm(A*x-b,Inf): %6.4f\ncvx_optval:      %6.4f\ncvx_status:      %s\n', norm(A*x-b,Inf), cvx_optval, cvx_status ) );
    disp( 'Optimal vector:' );
    disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
end
disp( sprintf( 'Residual vector; verify that the peaks match the objective (%6.4f):', cvx_optval ) );
disp( [ '   A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( ' ' );

try, input( 'Press Enter/Return for the next example...' ); clc; catch, end
echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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,[]);
    x_lp = xzz(1:n,:);
else,
    % linprog not present on this system
end

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

echo off

% Compare
if has_linprog,
    disp( sprintf( '\nResults:\n--------\nnorm(A*x_lp-b,1): %6.4f\nnorm(A*x-b,1): %6.4f\ncvx_optval: %6.4f\ncvx_status: %s\n', norm(A*x_lp-b,1), norm(A*x-b,1), cvx_optval, cvx_status ) );
    disp( 'Verify that x_lp == x:' );
    disp( [ '   x_lp  = [ ', sprintf( '%7.4f ', x_lp ), ']' ] );
    disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
else,
    disp( sprintf( '\nResults:\n--------\nnorm(A*x-b,1): %6.4f\ncvx_optval: %6.4f\ncvx_status: %s\n', norm(A*x-b,1), cvx_optval, cvx_status ) );
    disp( 'Optimal vector:' );
    disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
end
disp( 'Residual vector; verify the presence of several zero residuals:' );
disp( [ '   A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( ' ' );

try, input( 'Press Enter/Return for the next example...' ); clc; catch, end
echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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

% Compare
temp = sort(abs(A*x-b));
disp( sprintf( '\nResults:\n--------\nnorm_largest(A*x-b,k): %6.4f\ncvx_optval: %6.4f\ncvx_status: %s\n', norm_largest(A*x-b,k), cvx_optval, cvx_status ) );
disp( 'Optimal vector:' );
disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
disp( sprintf( 'Residual vector; verify a tie for %d-th place (%7.4f):', k, temp(end-k+1) ) );
disp( [ '   A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( ' ' );

try, input( 'Press Enter/Return for the next example...' ); clc; catch, end
echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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

% Compare
disp( sprintf( '\nResults:\n--------\nsum(huber(A*x-b)): %6.4f\ncvx_optval: %6.4f\ncvx_status: %s\n', sum(huber(A*x-b)), cvx_optval, cvx_status ) );
disp( 'Optimal vector:' );
disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
disp( 'Residual vector:' );
disp( [ '   A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( ' ' );

try, input( 'Press Enter/Return for the next example...' ); clc; catch, end
echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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

% Compare
disp( sprintf( '\nResults:\n--------\nnorm(A*x-b): %6.4f\ncvx_optval: %6.4f\ncvx_status: %s\n', norm(A*x-b), cvx_optval, cvx_status ) );
disp( 'Optimal vector:' );
disp( [ '   x     = [ ', sprintf( '%7.4f ', x ), ']' ] );
disp( 'Residual vector:' );
disp( [ '   A*x-b = [ ', sprintf( '%7.4f ', A*x-b ), ']' ] );
disp( 'Equality constraints:' );
disp( [ '   C*x   = [ ', sprintf( '%7.4f ', C*x ), ']' ] );
disp( [ '   d     = [ ', sprintf( '%7.4f ', d   ), ']' ] );

try, input( 'Press Enter/Return for the next example...' ); clc; catch, end
echo on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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
disp( ' ' );
disp( 'Generating tradeoff curve...' );
cvx_pause(false);
gamma = logspace( -2, 2, 20 );
l2norm = zeros(size(gamma));
l1norm = zeros(size(gamma));
fprintf( 1, '   gamma       norm(x,1)    norm(A*x-b)\n' );
fprintf( 1, '---------------------------------------\n' );
for k = 1:length(gamma),
    fprintf( 1, '%8.4e', gamma(k) );
    cvx_begin
        variable x(n);
        minimize( norm(A*x-b)+gamma(k)*norm(x,1) );
    cvx_end
    l1norm(k) = norm(x,1);
    l2norm(k) = norm(A*x-b);
    fprintf( 1, '   %8.4e   %8.4e\n', l1norm(k), l2norm(k) );
end
plot( l1norm, l2norm );
xlabel( 'norm(x,1)' );
ylabel( 'norm(A*x-b)' );
grid
disp( 'Done. (Check out the graph!)' );
randn( 'state', rnstate );
cvx_quiet(s_quiet);
cvx_pause(s_pause);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 ]
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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,
else,
    % quadprog not present on this system.
end

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

echo off

Results:
--------
norm(A*x-b): 4.6107
cvx_optval:  4.6107
cvx_status:  Solved

Verify that l <= x <= u:
   l     = [  0.3850 -0.2789 -0.9829 -0.9287 -3.0518 -0.0485  0.3182 -0.9417 ]
   x     = [  0.3850  0.6888 -0.1196 -0.8562  0.2923 -0.0485  0.4074 -0.6355 ]
   u     = [  0.7395  0.9403 -0.0290  1.6208  0.7450  2.4881  0.6919 -0.6355 ]
Residual vector:
   A*x-b = [  1.8604  0.2157 -0.2113  2.2664 -1.5320 -0.9598 -1.6815  0.2551  1.1079  0.5313  1.2390 -0.7555  0.2176 -0.3833  1.1950 -1.0823 ]
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.3: OTHER NORMS AND FUNCTIONS: INFINITY NORM %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if has_linprog,
else,
    % linprog not present on this system.
end

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

echo off

Results:
--------
norm(A*x-b,Inf): 0.7079
cvx_optval:      0.7079
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 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 ]
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SECTION 2.3: OTHER NORMS AND FUNCTIONS: ONE NORM %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if has_linprog,
else,
    % linprog not present on this system
end

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

echo off

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

Optimal vector:
   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 ]
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 ]
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 ]
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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): 7.9874
cvx_optval: 7.9874
cvx_status: Solved

Optimal vector:
   x     = [  0.5085  0.4243  0.8321 -0.5836  0.2810  0.9406  0.0218 -0.6629 ]
Residual vector:
   A*x-b = [  1.9546 -0.8458 -1.4935  3.9954 -2.5173  0.3568 -3.1377 -2.0091  1.2739  2.7589  2.5399  0.0131  0.8935 -1.4236 -0.4004 -1.4326 ]
Equality constraints:
   C*x   = [  1.2841 -0.7301 -1.4727 -1.5944 ]
   d     = [  1.2841 -0.7301 -1.4727 -1.5944 ]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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-02   3.3796e+00   2.0355e+00
1.6238e-02   3.3658e+00   2.0357e+00
2.6367e-02   3.3433e+00   2.0362e+00
4.2813e-02   3.3067e+00   2.0374e+00
6.9519e-02   3.2473e+00   2.0408e+00
1.1288e-01   3.1498e+00   2.0497e+00
1.8330e-01   2.9878e+00   2.0737e+00
2.9764e-01   2.7617e+00   2.1280e+00
4.8329e-01   2.3445e+00   2.2924e+00
7.8476e-01   1.4081e+00   2.8895e+00
1.2743e+00   3.7696e-01   3.9509e+00
2.0691e+00   1.0084e-08   4.4813e+00
3.3598e+00   2.8389e-11   4.4813e+00
5.4556e+00   5.7719e-11   4.4813e+00
8.8587e+00   1.4442e-11   4.4813e+00
1.4384e+01   4.6694e-10   4.4813e+00
2.3357e+01   1.7758e-11   4.4813e+00
3.7927e+01   9.1445e-13   4.4813e+00
6.1585e+01   3.3438e-12   4.4813e+00
1.0000e+02   2.1798e-14   4.4813e+00
Done. (Check out the graph!)