Hello everyone!

I want to implement the following minimization problem :

Where

**C is a NxL** DFT Matrix.

**Gamma and Theta** are constants.

**Epsilon is a positive weighting** factor between 0…1 and should provide a certain level of sparsity of b.

I choosed to rewrite the problem in CVX using epsilon as weighting factor for both problems:

```
cvx_begin
variable b(4) complex;
variable delta;
minimize(delta*(1-epsilon) + epsilon*(norm(b,1)));
subject to
(abs(imag(H_sy.'*(C*b))) - (real(H_sy.'*(C*b))-y)*tano) - delta <=0;
cvx_end
```

Where H_sy is just a Matrix, containing the h_k vectors with length N. In my example, L is choosen to 4.

Now, for some values of epsilon (in my case 0…0.3 ) CVX does not find a solution and reports the following:

```
Calling SDPT3 4.0: 14 variables, 1 equality constraints
------------------------------------------------------------
num. of constraints = 1
dim. of socp var = 14, num. of socp blk = 5
*******************************************************************
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|0.0e+00|3.2e+00|2.8e+01| 2.792711e+00 0.000000e+00| 0:0:00| chol * *
1|1.000|0.964|4.2e-17|2.1e-01|2.9e+00| 1.606495e+00 0.000000e+00| 0:0:00| chol * *
2|1.000|0.192|1.0e-15|1.7e-01|9.0e-01|-3.724179e+00 0.000000e+00| 0:0:00| chol 1 1
3|1.000|0.018|1.2e-14|1.7e-01|2.0e+01|-9.298855e+03 0.000000e+00| 0:0:00| chol 1 1
4|1.000|0.002|2.2e-13|1.7e-01|4.6e+05|-2.331268e+09 0.000000e+00| 0:0:00| chol 1 1
stop: primal infeas has deteriorated too much, 4.0e+00
5|1.000|0.000|2.2e-13|1.7e-01|4.6e+05|-2.331268e+09 0.000000e+00| 0:0:00|
prim_inf,dual_inf,relgap = 2.17e-13, 1.67e-01, 1.99e-04
sqlp stop: dual problem is suspected of being infeasible
-------------------------------------------------------------------
number of iterations = 5
residual of dual infeasibility
certificate X = 9.32e-23
reldist to infeas. <= 4.91e-24
Total CPU time (secs) = 0.10
CPU time per iteration = 0.02
termination code = 2
DIMACS: 2.2e-13 0.0e+00 2.0e-01 0.0e+00 -1.0e+00 2.0e-04
-------------------------------------------------------------------
------------------------------------------------------------
Status: Unbounded
Optimal value (cvx_optval): -Inf
```

So i have the following questions:

-I assume that my minimization problem is convex? … That holds true for every value of epsilon because epsilon is just a linear factor, right?

-Does the feasibility of the problem depend on the choice of epsilon?

-Or is there any other problem of my implementation?

Im a absolute beginner in CVX and convex problems - I apologize in case the answer is very obvious.

Best & thank you in advance!

Jul