% This script constructs a random equality-constrained norm minimization
% problem and solves it using CVX. You can also change p to +2 or +Inf
% to produce different results. Alternatively, you an replace
%     norm( A * x - b, p )
% with
%     norm_largest( A * x - b, 'largest', p )
% for 1 <= p <= 2 * n.

% Generate data
p = 1;
n = 10; m = 2*n; q=0.5*n;
A = randn(m,n);
b = randn(m,1);
C = randn(q,n);
d = randn(q,1);

% Create and solve problem
cvx_begin
   variable x(n)
   dual variable y
   minimize( norm( A * x - b, p ) )
   subject to
        y : C * x == d;
cvx_end

% Display results
disp( sprintf( 'norm(A*x-b,%g):', p ) );
disp( [ '   ans   =   ', sprintf( '%7.4f', norm(A*x-b,p) ) ] );
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   ), ']' ] );
disp( 'Lagrange multiplier for C*x==d:' );
disp( [ '   y     = [ ', sprintf( '%7.4f ', y ), ']' ] );

 
Calling SDPT3 4.0: 50 variables, 25 equality constraints
------------------------------------------------------------

 num. of constraints = 25
 dim. of socp   var  = 40,   num. of socp blk  = 20
 dim. of free   var  = 10 *** convert ublk to lblk
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
    NT      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|8.1e-01|2.8e+01|1.4e+04| 5.143640e+01  0.000000e+00| 0:0:00| chol  1  1 
 1|1.000|0.856|5.1e-06|4.1e+00|9.7e+02| 3.170797e+02  1.559244e+01| 0:0:00| chol  1  1 
 2|1.000|0.987|1.1e-06|6.3e-02|5.5e+01| 6.323422e+01  1.038030e+01| 0:0:00| chol  1  1 
 3|0.884|0.971|2.8e-06|2.7e-03|6.0e+00| 2.028692e+01  1.435501e+01| 0:0:00| chol  1  1 
 4|0.890|0.116|9.8e-06|3.3e-03|3.1e+00| 1.757201e+01  1.464034e+01| 0:0:00| chol  1  1 
 5|1.000|0.601|1.4e-06|1.3e-03|1.1e+00| 1.696414e+01  1.591961e+01| 0:0:00| chol  1  1 
 6|0.960|0.782|3.5e-07|2.9e-04|2.0e-01| 1.673312e+01  1.653658e+01| 0:0:00| chol  1  1 
 7|0.955|0.835|5.6e-08|4.8e-05|3.1e-02| 1.671466e+01  1.668403e+01| 0:0:00| chol  1  1 
 8|0.983|0.979|1.5e-08|1.0e-06|6.6e-04| 1.671341e+01  1.671276e+01| 0:0:00| chol  1  1 
 9|0.989|0.989|3.1e-09|4.5e-06|1.8e-05| 1.671339e+01  1.671338e+01| 0:0:00| chol  1  1 
10|1.000|0.989|1.8e-11|1.2e-07|4.3e-07| 1.671339e+01  1.671339e+01| 0:0:00| chol  1  1 
11|0.541|0.945|8.2e-12|2.9e-09|3.7e-08| 1.671339e+01  1.671339e+01| 0:0:00|
  stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
 number of iterations   = 11
 primal objective value =  1.67133912e+01
 dual   objective value =  1.67133911e+01
 gap := trace(XZ)       = 3.74e-08
 relative gap           = 1.09e-09
 actual relative gap    = 8.80e-10
 rel. primal infeas (scaled problem)   = 8.21e-12
 rel. dual     "        "       "      = 2.91e-09
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 7.0e+00, 1.0e+01, 6.0e+00
 norm(A), norm(b), norm(C) = 2.3e+01, 5.2e+00, 5.5e+00
 Total CPU time (secs)  = 0.28  
 CPU time per iteration = 0.03  
 termination code       =  0
 DIMACS: 1.5e-11  0.0e+00  8.0e-09  0.0e+00  8.8e-10  1.1e-09
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +16.7134
 
norm(A*x-b,1):
   ans   =   16.7134
Optimal vector:
   x     = [  0.0026 -0.1810 -0.4690 -0.1179 -0.0525  0.0758  0.0980  0.1892 -0.1700  0.4621 ]
Residual vector:
   A*x-b = [  0.0000 -2.0940  0.6100 -0.8864 -1.8009  0.0000 -0.9764 -0.7311  1.5586  1.1282 -1.4916 -0.0069 -0.0000 -0.0000 -1.5960  0.0629  0.5158  0.0000 -1.8174  1.4373 ]
Equality constraints:
   C*x   = [  0.1555  0.8186 -0.2926 -0.5408 -0.3086 ]
   d     = [  0.1555  0.8186 -0.2926 -0.5408 -0.3086 ]
Lagrange multiplier for C*x==d:
   y     = [ -6.7354  1.6456 -2.1811 -6.2263  0.4799 ]