Finding a point that satisfies many linear inequalities
m = 150;
n = 10;
seed = 0;
randn('state',seed);
A = randn(m,n);
b = randn(m,1);
fprintf(1, ['Starting with an infeasible set of %d inequalities ' ...
'in %d variables.\n'],m,n);
cvx_begin
variable x(n)
minimize( sum( max( A*x - b, 0 ) ) )
cvx_end
nv = length( find( A*x > b ) );
fprintf(1,'\nFound an x that violates %d out of %d inequalities.\n',nv,m);
Starting with an infeasible set of 150 inequalities in 10 variables.
Calling SDPT3: 310 variables, 150 equality constraints
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num. of constraints = 150
dim. of linear var = 300
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 mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|9.3e-01|2.5e+01|1.7e+05| 2.191997e+03| 0:0:00| chol 1 1
1|1.000|0.987|4.2e-06|6.2e-01|7.6e+03| 1.989611e+03| 0:0:00| chol 1 1
2|1.000|0.798|2.6e-07|1.5e-01|1.5e+03| 5.993105e+02| 0:0:00| chol 1 1
3|1.000|0.402|5.0e-07|9.0e-02|2.8e+02| 1.097681e+02| 0:0:00| chol 1 1
4|1.000|0.633|1.5e-06|3.3e-02|1.1e+02| 5.642133e+01| 0:0:00| chol 1 1
5|1.000|0.384|1.4e-07|2.0e-02|5.5e+01| 4.110139e+01| 0:0:00| chol 1 1
6|1.000|0.506|1.1e-07|1.0e-02|2.6e+01| 3.740113e+01| 0:0:00| chol 1 1
7|1.000|0.447|2.7e-08|5.6e-03|1.3e+01| 3.648738e+01| 0:0:00| chol 1 1
8|0.993|0.561|6.1e-09|2.4e-03|5.3e+00| 3.739303e+01| 0:0:00| chol 1 1
9|0.870|0.368|9.1e-10|1.5e-03|3.3e+00| 3.782964e+01| 0:0:00| chol 1 1
10|1.000|0.375|2.3e-10|9.7e-04|2.0e+00| 3.815325e+01| 0:0:00| chol 1 1
11|1.000|0.573|6.7e-11|4.1e-04|8.5e-01| 3.856006e+01| 0:0:01| chol 1 1
12|0.918|0.617|4.4e-12|1.6e-04|3.2e-01| 3.877476e+01| 0:0:01| chol 1 1
13|1.000|0.270|7.9e-12|1.2e-04|2.4e-01| 3.881145e+01| 0:0:01| chol 1 1
14|1.000|0.240|3.2e-12|8.8e-05|1.8e-01| 3.883697e+01| 0:0:01| chol 1 1
15|1.000|0.315|1.2e-12|6.0e-05|1.3e-01| 3.886172e+01| 0:0:01| chol 1 1
16|1.000|0.386|5.4e-13|3.7e-05|7.9e-02| 3.888229e+01| 0:0:01| chol 1 1
17|0.878|0.582|3.0e-13|1.5e-05|3.3e-02| 3.890202e+01| 0:0:01| chol 1 1
18|1.000|0.418|1.9e-13|9.0e-06|1.9e-02| 3.890799e+01| 0:0:01| chol 1 1
19|1.000|0.552|1.7e-13|1.1e-05|8.5e-03| 3.891278e+01| 0:0:01| chol 1 1
20|1.000|0.979|1.6e-14|4.5e-06|2.1e-04| 3.891668e+01| 0:0:01| chol 1 1
21|1.000|0.989|1.5e-15|1.1e-07|5.0e-06| 3.891676e+01| 0:0:01| chol 1 1
22|1.000|0.989|1.9e-15|2.6e-09|9.8e-08| 3.891676e+01| 0:0:01|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 22
primal objective value = 3.89167630e+01
dual objective value = 3.89167629e+01
gap := trace(XZ) = 9.81e-08
relative gap = 1.24e-09
actual relative gap = 1.08e-09
rel. primal infeas = 1.94e-15
rel. dual infeas = 2.61e-09
norm(X), norm(y), norm(Z) = 1.4e+01, 7.4e+00, 1.2e+01
norm(A), norm(b), norm(C) = 5.7e+01, 1.4e+01, 1.3e+01
Total CPU time (secs) = 1.0
CPU time per iteration = 0.0
termination code = 0
DIMACS: 6.8e-15 0.0e+00 1.7e-08 0.0e+00 1.1e-09 1.2e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +38.9168
Found an x that violates 57 out of 150 inequalities.