A convex optimization problem can be solved with SeDuMi and SDPT3, but not with Moesk

Now I have a conic optimization problem. Furthermore, I try to solve it by SeDuMi and SDPT3, where SeDuMi display is “Status: Inaccurate/Solved”, and SDPT3 return “Status: Solved”. However, when I use Mosek, the result returns “Status: Unbounded Optimal value (cvx_optval): -Inf”. Why is that?

It is my code, and run it requires loading the cvxquad package.

sig=[0.5,0;0,0.5];
X=[0,1;1,0];
Z=[1,0;0,-1];
I=[1,0;0,1];
Y=[0,-1i;1i,0];
a=0.764;
b=0.784;
n=2;
cvx_begin sdp 
     cvx_solver Mosek
    variable  rho(n,n) complex hermitian;
    cvx_precision best
    minimize quantum_rel_entr(rho,sig)
    %minimize quantum_rel_entr(rho,tau)
    subject to
        trace(rho) == 1;
        rho >= 0;
        real(trace(X*rho))>= a;
        real(trace(X*rho))<= b;
cvx_end
format long
cvx_optval
RelEnt=cvx_optval/log(2);

I duplicated your results. Both SDPT3 and SeDuMi resulted in cvx_optval = 0.330226.

The problem is that Mosek doesn’t “like” cvx_precision best, which should never be used with Mosek, and probably shouldn’t be used with other solvers. Upon removing cvx_precision best, Mosek produces the same solution as the other solvers. I’m not the Mosek or CVX developer, so don’t blame me for this behavior. Also note that the Inaccurate solved becomes just Solved when SeDuMi is used without cvx_precision best.

Here is the Mosek output. First with cvx_precision best. Then without.

With cvx_precision best:

Calling Mosek 9.3.6: 107 variables, 32 equality constraints
For improved efficiency, Mosek is solving the dual problem.

MOSEK Version 9.3.7 (Build date: 2021-10-11 10:42:47)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 32
Cones : 2
Scalar variables : 11
Matrix variables : 6
Integer variables : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.03
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 32
Cones : 2
Scalar variables : 11
Matrix variables : 6
Integer variables : 0

Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 27
Optimizer - Cones : 1
Optimizer - Scalar variables : 6 conic : 4
Optimizer - Semi-definite variables: 6 scalarized : 216
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 218 after factor : 218
Factor - dense dim. : 0 flops : 2.05e+04
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.4e+01 1.0e+00 1.7e+01 0.00e+00 1.600000000e+01 0.000000000e+00 1.0e+00 0.06
1 4.2e+00 2.9e-01 6.3e+00 -6.83e-01 3.210257708e+01 2.279937745e+01 2.9e-01 0.22
Optimizer terminated. Time: 0.31

Interior-point solution summary
Problem status : PRIMAL_INFEASIBLE
Solution status : PRIMAL_INFEASIBLE_CER
Dual. obj: 1.0279554867e+01 nrm: 7e-01 Viol. con: 0e+00 var: 1e-01 barvar: 5e-01 cones: 0e+00
Optimizer summary
Optimizer - time: 0.31
Interior-point - iterations : 1 time: 0.22
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00


Status: Unbounded
Optimal value (cvx_optval): -Inf

+++++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++++

Without cvx_precision best:

Calling Mosek 9.3.6: 107 variables, 32 equality constraints
For improved efficiency, Mosek is solving the dual problem.

MOSEK Version 9.3.7 (Build date: 2021-10-11 10:42:47)
Copyright © MOSEK ApS, Denmark. WWW: mosek.com
Platform: Windows/64-X86

Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 32
Cones : 2
Scalar variables : 11
Matrix variables : 6
Integer variables : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.02
Lin. dep. - number : 0
Presolve terminated. Time: 0.05
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 32
Cones : 2
Scalar variables : 11
Matrix variables : 6
Integer variables : 0

Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 27
Optimizer - Cones : 1
Optimizer - Scalar variables : 6 conic : 4
Optimizer - Semi-definite variables: 6 scalarized : 216
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 218 after factor : 218
Factor - dense dim. : 0 flops : 2.05e+04
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.4e+01 1.0e+00 1.7e+01 0.00e+00 1.600000000e+01 0.000000000e+00 1.0e+00 0.08
1 4.2e+00 2.9e-01 6.3e+00 -6.83e-01 3.210257708e+01 2.279937745e+01 2.9e-01 0.19
2 2.2e+00 1.5e-01 2.7e+00 3.53e-01 1.371965388e+01 7.917115553e+00 1.5e-01 0.19
3 4.7e-01 3.3e-02 3.2e-01 5.29e-01 -2.634668838e+00 -4.121972995e+00 3.3e-02 0.20
4 9.2e-02 6.4e-03 2.7e-02 1.01e+00 -7.523690680e+00 -7.814658111e+00 6.4e-03 0.22
5 1.4e-02 1.0e-03 1.7e-03 1.01e+00 -9.010308909e+00 -9.055367493e+00 1.0e-03 0.23
6 3.3e-03 2.3e-04 1.8e-04 1.03e+00 -9.204268084e+00 -9.214332462e+00 2.3e-04 0.25
7 3.4e-04 2.3e-05 6.1e-06 9.99e-01 -9.246668743e+00 -9.247698534e+00 2.3e-05 0.25
8 2.2e-06 1.5e-07 3.2e-09 1.00e+00 -9.252630201e+00 -9.252636855e+00 1.5e-07 0.25
9 6.2e-08 4.3e-09 1.5e-11 1.00e+00 -9.252670172e+00 -9.252670362e+00 4.3e-09 0.27
10 3.2e-09 9.6e-10 1.8e-13 1.00e+00 -9.252671420e+00 -9.252671430e+00 2.2e-10 0.27
Optimizer terminated. Time: 0.34

Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -9.2526714204e+00 nrm: 1e+01 Viol. con: 5e-09 var: 0e+00 barvar: 0e+00 cones: 0e+00
Dual. obj: -9.2526714303e+00 nrm: 1e+00 Viol. con: 0e+00 var: 6e-10 barvar: 3e-09 cones: 0e+00
Optimizer summary
Optimizer - time: 0.34
Interior-point - iterations : 10 time: 0.27
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00


Status: Solved
Optimal value (cvx_optval): +0.330226

Thanks for contributing! Your answer solved my confusion.