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<h1>Equality constrained norm minimization.</h1>
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<pre class="codeinput">
<span class="comment">% This script constructs a random equality-constrained norm minimization</span>
<span class="comment">% problem and solves it using CVX. You can also change p to +2 or +Inf</span>
<span class="comment">% to produce different results. Alternatively, you an replace</span>
<span class="comment">% norm( A * x - b, p )</span>
<span class="comment">% with</span>
<span class="comment">% norm_largest( A * x - b, 'largest', p )</span>
<span class="comment">% for 1 &lt;= p &lt;= 2 * n.</span>
<span class="comment">% Generate data</span>
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);
<span class="comment">% Create and solve problem</span>
cvx_begin
variable <span class="string">x(n)</span>
dual <span class="string">variable</span> <span class="string">y</span>
minimize( norm( A * x - b, p ) )
subject <span class="string">to</span>
y : C * x == d;
cvx_end
<span class="comment">% Display results</span>
disp( sprintf( <span class="string">'norm(A*x-b,%g):'</span>, p ) );
disp( [ <span class="string">' ans = '</span>, sprintf( <span class="string">'%7.4f'</span>, norm(A*x-b,p) ) ] );
disp( <span class="string">'Optimal vector:'</span> );
disp( [ <span class="string">' x = [ '</span>, sprintf( <span class="string">'%7.4f '</span>, x ), <span class="string">']'</span> ] );
disp( <span class="string">'Residual vector:'</span> );
disp( [ <span class="string">' A*x-b = [ '</span>, sprintf( <span class="string">'%7.4f '</span>, A*x-b ), <span class="string">']'</span> ] );
disp( <span class="string">'Equality constraints:'</span> );
disp( [ <span class="string">' C*x = [ '</span>, sprintf( <span class="string">'%7.4f '</span>, C*x ), <span class="string">']'</span> ] );
disp( [ <span class="string">' d = [ '</span>, sprintf( <span class="string">'%7.4f '</span>, d ), <span class="string">']'</span> ] );
disp( <span class="string">'Lagrange multiplier for C*x==d:'</span> );
disp( [ <span class="string">' y = [ '</span>, sprintf( <span class="string">'%7.4f '</span>, y ), <span class="string">']'</span> ] );
</pre>
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<pre class="codeoutput">
Calling SDPT3: 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|8.3e-01|2.9e+01|1.5e+04| 8.471540e+01 0.000000e+00| 0:0:00| chol 1 1
1|1.000|0.964|3.1e-06|1.1e+00|2.8e+02| 1.464735e+02 1.231256e+01| 0:0:00| chol 1 1
2|0.961|0.990|6.9e-07|2.0e-02|1.0e+01| 2.225613e+01 1.265515e+01| 0:0:00| chol 1 1
3|0.935|0.790|2.0e-05|5.0e-03|1.8e+00| 1.789246e+01 1.620084e+01| 0:0:00| chol 1 1
4|0.862|0.455|2.3e-05|2.8e-03|7.7e-01| 1.736289e+01 1.665120e+01| 0:0:00| chol 1 1
5|1.000|0.442|1.3e-07|1.6e-03|3.8e-01| 1.725395e+01 1.690178e+01| 0:0:00| chol 1 1
6|1.000|0.824|3.0e-08|2.7e-04|6.4e-02| 1.722824e+01 1.716909e+01| 0:0:00| chol 1 1
7|0.988|0.777|5.9e-09|6.1e-05|1.4e-02| 1.722561e+01 1.721288e+01| 0:0:00| chol 1 1
8|1.000|0.283|1.3e-09|4.4e-05|9.4e-03| 1.722491e+01 1.721622e+01| 0:0:00| chol 1 1
9|1.000|0.280|4.7e-10|3.1e-05|6.7e-03| 1.722479e+01 1.721858e+01| 0:0:00| chol 1 1
10|1.000|0.688|1.7e-10|9.8e-06|2.1e-03| 1.722470e+01 1.722275e+01| 0:0:00| chol 1 1
11|0.993|0.866|4.6e-11|1.4e-05|3.7e-04| 1.722465e+01 1.722439e+01| 0:0:00| chol 1 1
12|0.988|0.988|1.9e-12|2.5e-06|2.0e-05| 1.722465e+01 1.722465e+01| 0:0:00| chol 1 1
13|1.000|0.944|1.5e-13|1.4e-07|1.2e-06| 1.722465e+01 1.722465e+01| 0:0:00| chol 1 1
14|0.575|0.944|6.6e-14|8.1e-09|9.7e-08| 1.722465e+01 1.722465e+01| 0:0:00|
stop: max(relative gap, infeasibilities) &lt; 1.49e-08
-------------------------------------------------------------------
number of iterations = 14
primal objective value = 1.72246500e+01
dual objective value = 1.72246500e+01
gap := trace(XZ) = 9.66e-08
relative gap = 2.72e-09
actual relative gap = 1.18e-09
rel. primal infeas = 6.57e-14
rel. dual infeas = 8.07e-09
norm(X), norm(y), norm(Z) = 7.5e+00, 6.8e+00, 6.2e+00
norm(A), norm(b), norm(C) = 2.4e+01, 5.8e+00, 5.5e+00
Total CPU time (secs) = 0.17
CPU time per iteration = 0.01
termination code = 0
DIMACS: 1.3e-13 0.0e+00 2.2e-08 0.0e+00 1.2e-09 2.7e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +17.2247
norm(A*x-b,1):
ans = 17.2246
Optimal vector:
x = [ -0.5253 -0.6745 0.3881 -0.3432 0.6278 -0.7627 0.0400 0.6648 1.0041 -0.4625 ]
Residual vector:
A*x-b = [ 0.0000 -0.0000 -0.8708 -1.0281 -0.0000 1.5043 0.0000 1.3248 0.2570 -2.1089 2.3600 0.7021 -1.2957 0.9965 0.1152 0.8386 -2.2360 0.0000 -1.1761 -0.4105 ]
Equality constraints:
C*x = [ -0.0798 0.7377 1.5352 1.2677 1.9951 ]
d = [ -0.0798 0.7377 1.5352 1.2677 1.9951 ]
Lagrange multiplier for C*x==d:
y = [ 2.2526 0.2934 1.6808 -0.2579 4.4457 ]
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