Dear all,

I’m solving the following optimization problem with CVX:

```
cvx_begin quiet
variable W(N,N) hermitian
variable t
maximize( trace(R(:,:,1)*W) )
subject to
for i=1:N
W(i,i) == t;
end
trace(J(:,:,1)*W) == 1;
W == hermitian_semidefinite(N);
cvx_end
```

Where R(:,:,1) and J(:,:,1) are Hermitian semidefinite positive matrices. Depending the value of these matrices, I obtain the following result:

## Calling SDPT3 4.0: 1296 variables, 36 equality constraints

num. of constraints = 36

dim. of sdp var = 72, num. of sdp blk = 1

SDPT3: Infeasible path-following algorithms

## version predcorr gam expon scale_data

HKM 1 0.000 1 0

it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime

## 0|0.000|0.000|3.6e+01|1.0e+01|5.2e+04|-7.440721e+02 0.000000e+00| 0:0:00| chol 1 1

1|1.000|0.969|2.3e-05|6.1e-01|2.3e+03|-7.255549e+02 -1.849722e+01| 0:0:00| chol 1 1

2|1.000|0.254|9.6e-05|4.8e-01|3.0e+03|-4.317215e+03 -2.617684e+01| 0:0:00| chol 1 1

3|0.016|0.016|1.1e-04|4.7e-01|3.6e+03|-1.885771e+04 -2.850937e+01| 0:0:00| chol 1 1

4|0.009|0.013|1.2e-04|4.7e-01|6.0e+03|-1.070550e+05 -3.634638e+01| 0:0:00| chol 1 1

5|0.008|0.012|1.2e-04|4.6e-01|2.3e+04|-1.021671e+06 -8.827492e+01| 0:0:00| chol 1 2

6|0.001|0.002|1.2e-04|4.6e-01|7.2e+04|-3.831836e+06 -1.840813e+02| 0:0:00| chol 2 2

7|0.001|0.002|1.2e-04|4.6e-01|3.1e+05|-2.008218e+07 -4.546200e+02| 0:0:00| chol 2 2

8|0.000|0.000|1.2e-04|4.6e-01|4.5e+05|-2.972326e+07 -5.995847e+02| 0:0:00| chol 2 2

9|0.000|0.000|1.2e-04|4.6e-01|6.5e+05|-4.249214e+07 -7.230162e+02| 0:0:00| chol 2 2

10|0.000|0.000|1.2e-04|4.6e-01|7.4e+05|-4.726870e+07 -7.231650e+02| 0:0:00| chol 2 * 4

11|0.000|0.000|1.1e-04|4.6e-01|7.5e+05|-4.537039e+07 -6.132804e+02| 0:0:00| chol 2 3

12|0.000|0.000|1.1e-04|4.6e-01|8.2e+05|-4.905174e+07 -4.493363e+02| 0:0:00| chol 2 2

13|0.000|0.000|1.1e-04|4.6e-01|9.4e+05|-5.479629e+07 -5.572144e+02| 0:0:00| chol 2 2

14|0.000|0.000|1.1e-04|4.6e-01|1.1e+06|-7.018672e+07 -7.939857e+02| 0:0:00| chol 2 2

15|0.000|0.000|1.1e-04|4.6e-01|1.3e+06|-8.551341e+07 -1.010778e+03| 0:0:00| chol 2 3

16|0.000|0.000|1.1e-04|4.6e-01|1.8e+06|-1.254625e+08 -1.347171e+03| 0:0:00| chol 3 3

17|0.000|0.000|1.1e-04|4.6e-01|2.7e+06|-1.973905e+08 -1.649807e+03| 0:0:00| chol 3 3

18|0.000|0.000|1.3e-04|4.6e-01|4.2e+06|-3.216027e+08 -2.165831e+03| 0:0:00| chol 3 3

19|0.000|0.000|1.6e-04|4.6e-01|5.5e+06|-4.286403e+08 -2.621341e+03| 0:0:00| chol 3 3

20|0.000|0.000|2.5e-04|4.6e-01|7.1e+06|-5.696079e+08 -3.248557e+03| 0:0:00| chol 3 3

21|0.000|0.000|2.3e-04|4.6e-01|8.4e+06|-6.875518e+08 -3.765532e+03| 0:0:00| chol 3 3

22|0.000|0.000|3.3e-04|4.6e-01|1.2e+07|-1.003153e+09 -4.627078e+03| 0:0:00| chol 3 * 5

23|0.000|0.000|3.5e-04|4.6e-01|1.5e+07|-1.314203e+09 -5.184290e+03| 0:0:00| chol 3 * 5

24|0.000|0.000|7.9e-04|4.6e-01|2.0e+07|-1.835452e+09 -6.464178e+03| 0:0:00| chol 3 * 5

25|0.000|0.000|8.4e-04|4.6e-01|2.6e+07|-2.312570e+09 -7.098093e+03| 0:0:00| chol 3 * 6

26|0.000|0.000|1.9e-03|4.6e-01|3.4e+07|-3.217746e+09 -9.815672e+03| 0:0:00| chol 3 * 5

27|0.000|0.000|2.2e-03|4.6e-01|3.8e+07|-3.609259e+09 -1.040805e+04| 0:0:00| chol * 5 * 5

28|0.000|0.001|2.4e-03|4.6e-01|3.9e+07|-4.012248e+09 -1.394667e+04| 0:0:00| chol 4 * 6

29|0.000|0.000|2.3e-03|4.6e-01|4.2e+07|-4.271910e+09 -1.518461e+04| 0:0:00| chol 4 * 6

30|0.000|0.000|4.1e-03|4.6e-01|5.1e+07|-5.394488e+09 -1.831299e+04| 0:0:00| chol 4 * 5

31|0.000|0.000|4.0e-03|4.5e-01|5.4e+07|-5.822215e+09 -1.936435e+04| 0:0:00|

sqlp stop: dual problem is suspected of being infeasible

## number of iterations = 31

residual of dual infeasibility

certificate X = 1.71e-10

reldist to infeas. <= 2.59e-11

Total CPU time (secs) = 0.45

CPU time per iteration = 0.01

termination code = 2

DIMACS: 4.0e-03 0.0e+00 6.2e-01 0.0e+00 -1.0e+00 9.3e-03

Status: Unbounded

Optimal value (cvx_optval): +Inf

However, to the best of my knowledge, the problem is not unbounded or unfeasible.

Could anyone help on how to identify the problem?

Thank you,