Cvx can run without cvx_solver mosek, but cannot run with solver mosek

1

Hi, I have a question and hope to get your help. Thank you.
This is code without cvx_solver mosek.

P=1;
M=8;
G=[0.000170199312934811 + 6.06577565666215e-05i,-0.000305124492145413 + 0.000216991512142075i,0.000284631876775653 - 0.000113931157972261i,-0.000158700432294316 + 9.12280586319212e-05i,0.000100175288175321 - 0.000212838119149561i,6.94790851279640e-05 + 0.000288327306170263i,-8.23893382719307e-05 - 1.26326105903413e-05i,3.18052566221984e-05 + 0.000214636098843472i;0.000260385237916695 + 0.000112482539175534i,-6.47027410656752e-05 + 7.36246979171719e-05i,0.000384838227832422 + 3.61270779933175e-05i,-0.000110356495807971 + 0.000138308494069889i,0.000206014182594723 - 0.000192072540435726i,-9.30303504529306e-05 + 0.000194631340490199i,2.45467219391394e-05 - 0.000231281896138424i,0.000238435278455976 + 0.000296829157249881i;0.000101605035742007 + 0.000135308395453357i,-0.000151922199843739 - 5.21931766098463e-06i,0.000337295546395561 - 9.69972466522999e-05i,-9.81797902321573e-05 + 0.000202256325414545i,0.000132771653735296 - 7.02106229024501e-05i,-0.000232516191403371 + 0.000224869501527691i,3.78092590880102e-05 - 0.000294662081789046i,-4.08207616522847e-05 + 0.000314884062104210i;7.84264532876304e-05 + 6.55426072236379e-05i,-0.000355011097928199 - 0.000151895703483664i,0.000264045782429546 + 0.000153607681353190i,-0.000275280758325934 + 9.53129097417783e-05i,0.000283028276730451 - 7.62489185150245e-05i,-0.000149656020931544 + 0.000239657553613363i,0.000158909468802005 - 0.000259707160160399i,4.70844621038254e-05 + 0.000177945221668646i;0.000211586287949353 + 0.000152728738425378i,-0.000133051072523730 - 8.28075496462411e-05i,0.000353764385722472 + 6.08326589428266e-05i,-0.000316284811560485 + 4.02995175797813e-05i,0.000285671184679029 - 8.38005605328197e-06i,-0.000295211113603451 + 0.000108880691862089i,1.02512277810118e-05 - 0.000250410092005423i,-0.000181365024760836 + 0.000100308730203992i;-1.45259024048908e-05 + 0.000254200846074134i,-0.000293089436423124 - 0.000294318702373552i,8.65957301298470e-05 + 6.98008487396994e-05i,-0.000255671062575227 - 7.83022380195911e-05i,0.000223738695755060 + 8.92686523938305e-06i,-4.23698265385615e-05 + 2.47342720498652e-05i,0.000272696901218335 - 0.000121045296319604i,-0.000153867509446520 + 0.000104005401713423i;5.27315098219125e-05 + 0.000200407152438837i,-0.000229141216445705 - 0.000138444161483631i,0.000230645197698344 + 0.000277226768764386i,-0.000168924776580887 - 0.000336829896417031i,0.000380088393173370 + 8.66147076522055e-08i,-0.000191856887814606 + 0.000192098308980634i,9.61099632599656e-05 - 0.000128780669165083i,-6.13725877504738e-05 + 8.41896452879896e-05i;0.000126225080999300 + 0.000209624033385059i,4.90553742259838e-05 - 0.000172453588207806i,0.000122681884695804 + 4.23215720662085e-05i,-0.000212727071419110 - 0.000261036425522721i,9.00365504796512e-05 + 0.000111499745659919i,-0.000389495988243510 - 7.10813185451640e-05i,0.000151755919678257 - 8.90180778248533e-05i,-0.000252141256987222 + 6.80588932815459e-05i];
la=1e7;
cvx_begin sdp
    variable W(M,M) complex semidefinite
    variable R0(M,M) complex semidefinite
    expression CRB
    CRB=real(trace(inv_pos(real( G*(W+R0)'*G'))))
    obj1=CRB/la;
    minimize obj1
    subject to 
    trace(W+R0)<=P;
    W>=0;
    R0>=0;
cvx_end

The result is Status: Solved
Optimal value (cvx_optval): +1.7881

This is code with cvx_solver mosek.

cvx_begin sdp
cvx_solver mosek
    variable W(M,M) complex semidefinite
    variable R0(M,M) complex semidefinite
    expression CRB
    CRB=real(trace(inv_pos(real( G*(W+R0)'*G'))));
    obj1=CRB/la;
    minimize obj1
    subject to 
    trace(W+R0)<=P;
    W>=0;
    R0>=0;
cvx_end

The result is ------------------------------------------------------------
Status: Infeasible
Optimal value (cvx_optval): +Inf

2

I presume your without cvx_solver mosek means you used SeDuMi or SDPT3, whichever is set as your default solver.

I ran this problem with 3 solvers with the following results. CVX/Solver logs are below.

Status:
Mosek 10.1.12: infeasible
SeDuMi: Inaccurate Solved with optimal objective value = 1.91091
SDPT3: Solved with optimal objective value = 1.7881

The input data doesn’t look too wonderful, so perhaps poor numerical scaling of the input data is contributing to difficulties However, the input data magnitude isn’t small enough to trigger any Mosek warnings. I will defer to Mosek personnel to further assess.

Calling Mosek 10.1.10: 449 variables, 192 equality constraints
For improved efficiency, Mosek is solving the dual problem.

MOSEK Version 10.1.12 (Build date: 2023-9-20 10:51:32)
Copyright (c) MOSEK ApS, Denmark WWW: mosek.com
Platform: Windows/64-X86

Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 192
Affine conic cons. : 0
Disjunctive cons. : 0
Cones : 64
Scalar variables : 193
Matrix variables : 4 (scalarized: 544)
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. - primal attempts : 1 successes : 1
Lin. dep. - dual attempts : 0 successes : 0
Lin. dep. - primal deps. : 0 dual deps. : 0
Presolve terminated. Time: 0.00
GP based matrix reordering started.
GP based matrix reordering terminated.
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 128
Optimizer - Cones : 64
Optimizer - Scalar variables : 193 conic : 192
Optimizer - Semi-definite variables: 4 scalarized : 544
Factor - setup time : 0.03
Factor - dense det. time : 0.00 GP order time : 0.02
Factor - nonzeros before factor : 8256 after factor : 8256
Factor - dense dim. : 0 flops : 3.61e+06
Factor - GP saved nzs : 0 GP saved flops : 1.02e+06
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 4.0e+00 2.0e+00 1.0e+00 0.00e+00 0.000000000e+00 -4.000000000e-07 1.0e+00 0.05
1 1.2e+00 6.2e-01 5.3e-01 -9.66e-01 -2.039008599e+00 -1.234217737e-06 3.1e-01 0.06
2 1.0e-02 5.1e-03 1.6e-02 -8.87e-01 -3.982519627e+01 -1.980308313e-05 2.5e-03 0.08
3 1.7e-03 8.5e-04 2.1e-03 2.55e-01 -2.428681165e+01 -4.869996816e-05 4.2e-04 0.08
4 3.2e-04 1.6e-04 3.6e-04 1.77e-01 -1.997173621e+01 -1.234062260e-04 7.9e-05 0.09
5 5.2e-05 2.6e-05 5.4e-05 8.63e-02 -1.759433189e+01 -3.127953359e-04 1.3e-05 0.09
6 8.0e-06 4.0e-06 7.8e-06 3.55e-02 -1.534215360e+01 -7.942922860e-04 2.0e-06 0.09
7 1.9e-06 1.1e-06 1.4e-06 9.54e-02 -8.831356510e+00 -1.893676516e-03 4.7e-07 0.11
8 6.4e-07 3.7e-07 6.5e-07 -2.70e-01 -1.687802116e+01 -2.959952057e-03 1.6e-07 0.11
9 1.5e-07 8.4e-08 1.1e-07 6.29e-02 -8.522801920e+00 -5.707126299e-03 3.6e-08 0.11
10 3.3e-08 1.9e-08 2.2e-08 2.33e-02 -7.339925943e+00 -1.258274035e-02 8.2e-09 0.12
11 6.5e-09 4.0e-09 4.4e-09 -5.16e-03 -7.347632580e+00 -2.795713603e-02 1.6e-09 0.12
12 1.4e-09 2.7e-09 9.4e-10 -1.55e-02 -7.158348801e+00 -6.032933661e-02 3.5e-10 0.12
13 3.4e-10 5.8e-09 2.2e-10 -7.60e-05 -6.656943204e+00 -1.273002582e-01 8.5e-11 0.14
14 7.4e-11 9.4e-09 4.5e-11 9.08e-03 -6.260397464e+00 -2.734293802e-01 1.8e-11 0.14
15 1.6e-11 2.6e-08 9.3e-12 1.05e-02 -6.057008416e+00 -5.877320571e-01 4.0e-12 0.16
16 3.6e-12 5.9e-08 2.0e-12 2.38e-02 -6.061123132e+00 -1.206431994e+00 9.3e-13 0.16
17 1.9e-12 3.2e-08 1.0e-12 6.30e-02 -6.170931938e+00 -1.703099624e+00 5.0e-13 0.17
18 3.9e-13 1.1e-08 1.6e-13 1.51e-01 -5.728607604e+00 -3.087399912e+00 1.0e-13 0.17
19 3.9e-13 1.1e-08 1.6e-13 1.51e-01 -5.728607604e+00 -3.087399912e+00 1.0e-13 0.19
20 3.9e-13 1.1e-08 1.6e-13 1.51e-01 -5.728607604e+00 -3.087399912e+00 1.0e-13 0.20
Optimizer terminated. Time: 0.22

Interior-point solution summary
Problem status : DUAL_INFEASIBLE
Solution status : DUAL_INFEASIBLE_CER
Primal. obj: -1.4764019033e-06 nrm: 2e+00 Viol. con: 4e-13 var: 0e+00 barvar: 0e+00 cones: 7e-14
Optimizer summary
Optimizer - time: 0.22
Interior-point - iterations : 21 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: Infeasible
Optimal value (cvx_optval): +Inf

Calling SeDuMi 1.3.4: 449 variables, 192 equality constraints
For improved efficiency, SeDuMi is solving the dual problem.

SeDuMi 1.3.4 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 192, order n = 162, dim = 706, blocks = 69
nnz(A) = 16784 + 0, nnz(ADA) = 32832, nnz(L) = 16512
it : by gap delta rate t/tP t/tD* feas cg cg prec
0 : 1.92E-01 0.000
1 : -1.60E-05 9.43E-03 0.000 0.0491 0.9900 0.9900 -0.89 1 1 2.8E+00
2 : -4.84E-05 6.14E-05 0.000 0.0065 0.9990 0.9990 -0.19 1 1 5.1E-02
3 : -9.87E-05 1.64E-05 0.000 0.2665 0.9000 0.9000 0.10 1 1 2.5E-02
4 : -1.97E-04 4.34E-06 0.000 0.2654 0.9000 0.9000 0.06 1 1 1.3E-02
5 : -3.89E-04 1.15E-06 0.000 0.2651 0.9000 0.9000 0.04 1 1 6.4E-03
6 : -7.62E-04 3.05E-07 0.000 0.2650 0.9000 0.9000 0.02 1 1 3.2E-03
7 : -1.49E-03 8.08E-08 0.000 0.2650 0.9000 0.9000 0.01 1 1 1.7E-03
8 : -2.89E-03 2.14E-08 0.000 0.2650 0.9000 0.9000 0.01 1 1 8.5E-04
9 : -5.62E-03 5.68E-09 0.000 0.2651 0.9000 0.9000 0.00 1 1 4.4E-04
10 : -1.09E-02 1.51E-09 0.000 0.2651 0.9000 0.9000 0.00 1 1 2.3E-04
11 : -2.12E-02 3.99E-10 0.000 0.2651 0.9000 0.9000 0.00 1 1 1.2E-04
12 : -4.12E-02 1.06E-10 0.000 0.2651 0.9000 0.9000 0.00 1 1 6.0E-05
13 : -7.98E-02 2.81E-11 0.000 0.2652 0.9000 0.9000 0.00 2 2 3.1E-05
14 : -1.54E-01 7.45E-12 0.000 0.2655 0.9000 0.9000 0.00 2 2 1.6E-05
15 : -2.98E-01 1.98E-12 0.000 0.2658 0.9000 0.9000 0.00 2 2 8.2E-06
16 : -5.69E-01 5.29E-13 0.000 0.2670 0.9000 0.9000 0.01 2 2 4.2E-06
17 : -1.07E+00 1.43E-13 0.000 0.2706 0.9000 0.9000 0.01 4 3 2.2E-06
18 : -1.91E+00 4.04E-14 0.000 0.2822 0.9000 0.9000 0.04 5 5 1.1E-06
Run into numerical problems.

iter seconds digits cx by
18 2.3 Inf -1.5174078094e+01 -1.9109149727e+00
|Ax-b| = 2.4e-06, [Ay-c]_+ = 5.2E-07, |x|= 9.4e+06, |y|= 2.1e+07

Detailed timing (sec)
Pre IPM Post
2.443E+00 4.968E+00 4.000E-02
Max-norms: ||b||=1.000000e-07, ||c|| = 2,
Cholesky |add|=3, |skip| = 0, ||L.L|| = 1.07377e+06.

Status: Inaccurate/Solved
Optimal value (cvx_optval): +1.91091

Calling SDPT3 4.0: 449 variables, 192 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.

num. of constraints = 192
dim. of sdp var = 192, num. of sdp blk = 68
dim. of linear var = 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|1.2e+02|1.1e+01|2.3e+04| 1.000000e+01 0.000000e+00| 0:0:01| chol 1 1
1|0.924|0.860|9.0e+00|1.7e+00|4.1e+03|-7.608139e+01 -1.076639e-05| 0:0:02| chol 1 1
2|0.910|0.949|8.1e-01|9.4e-02|1.9e+02|-8.058108e+01 -1.384681e-05| 0:0:02| chol 1 1
3|0.793|0.661|1.7e-01|3.3e-02|7.3e+01|-4.184133e+01 -3.029635e-05| 0:0:02| chol 1 1
4|0.901|0.668|1.7e-02|1.1e-02|6.3e+01| 2.364067e+01 -9.458752e-05| 0:0:02| chol 1 1
5|0.034|0.809|1.6e-02|2.8e-03|6.5e+01|-3.497308e+01 -6.892460e-04| 0:0:03| chol 1 1
6|0.854|0.636|2.4e-03|1.1e-03|1.6e+01|-1.496595e+01 -9.670838e-04| 0:0:03| chol 1 1
7|0.485|0.553|1.3e-03|5.3e-04|8.8e+00|-2.449019e+01 -1.884930e-03| 0:0:03| chol 1 1
8|0.666|0.471|4.5e-04|2.9e-04|5.3e+00|-2.064468e+01 -3.314544e-03| 0:0:03| chol 1 1
9|0.931|0.571|4.4e-05|1.4e-04|1.2e+01| 2.919214e+00 -7.772938e-03| 0:0:03| chol 1 1
10|0.060|0.503|4.1e-05|7.5e-05|1.0e+01|-1.217376e+01 -2.832487e-02| 0:0:03| chol 1 1
11|0.823|0.722|7.3e-06|2.2e-05|2.4e+00|-5.061120e+00 -5.047951e-02| 0:0:03| chol 1 1
12|0.631|0.573|2.7e-06|9.8e-06|1.3e+00|-4.392879e+00 -1.012338e-01| 0:0:03| chol 1 1
13|0.630|0.504|9.9e-07|5.0e-06|1.1e+00|-3.679376e+00 -1.887294e-01| 0:0:03| chol 1 1
14|0.410|0.580|5.8e-07|2.3e-06|1.4e+00|-2.954769e+00 -3.891815e-01| 0:0:03| chol 1 1
15|0.330|0.423|3.9e-07|1.4e-06|1.3e+00|-3.368128e+00 -5.438536e-01| 0:0:03| chol 1 1
16|0.357|0.429|2.5e-07|8.6e-07|1.3e+00|-3.480252e+00 -7.530662e-01| 0:0:03| chol 1 1
17|0.434|0.453|1.4e-07|5.0e-07|1.2e+00|-3.218879e+00 -1.025601e+00| 0:0:03| chol 1 1
18|0.508|0.495|7.0e-08|2.8e-07|1.0e+00|-2.692875e+00 -1.316457e+00| 0:0:03| chol 1 1
19|0.611|0.538|2.7e-08|1.4e-07|7.6e-01|-2.174660e+00 -1.569655e+00| 0:0:03| chol 1 1
20|0.795|0.553|5.6e-09|6.9e-08|4.9e-01|-1.778489e+00 -1.735356e+00| 0:0:03| chol 1 1
21|0.908|0.640|5.1e-10|2.6e-08|2.9e-01|-1.684059e+00 -1.821321e+00| 0:0:03| chol 1 1
22|0.468|1.000|2.7e-10|2.4e-09|1.6e-01|-1.717460e+00 -1.837510e+00| 0:0:03| chol 1 1
23|0.760|0.624|6.6e-11|3.1e-10|8.4e-02|-1.752561e+00 -1.821819e+00| 0:0:03| chol 1 1
24|0.720|0.478|1.8e-11|6.1e-10|5.4e-02|-1.766295e+00 -1.813553e+00| 0:0:03| chol 1 1
25|0.736|0.330|4.8e-12|6.0e-10|3.5e-02|-1.775728e+00 -1.807941e+00| 0:0:03| chol 1 1
26|0.833|0.305|8.3e-13|2.6e-11|2.3e-02|-1.781530e+00 -1.803271e+00| 0:0:04| chol 1 1
27|0.575|0.698|3.6e-13|8.8e-12|1.2e-02|-1.783832e+00 -1.794628e+00| 0:0:04| chol 1 1
28|0.893|0.308|4.8e-14|6.9e-12|8.1e-03|-1.785432e+00 -1.793268e+00| 0:0:04| chol 1 1
29|0.416|1.000|3.3e-14|8.2e-13|3.4e-03|-1.786233e+00 -1.789056e+00| 0:0:04| chol 1 1
30|0.869|0.669|5.3e-14|1.1e-12|1.7e-03|-1.787257e+00 -1.788848e+00| 0:0:04| chol 1 1
31|0.494|1.000|2.5e-14|8.2e-13|9.0e-04|-1.787583e+00 -1.788319e+00| 0:0:04| chol 1 1
32|0.679|0.923|1.5e-14|8.8e-13|5.6e-04|-1.787818e+00 -1.788280e+00| 0:0:04| chol 1 1
33|0.739|1.000|1.3e-14|8.2e-13|2.7e-04|-1.787966e+00 -1.788190e+00| 0:0:04| chol 1 1
34|0.671|0.912|9.7e-15|8.9e-13|1.6e-04|-1.788026e+00 -1.788151e+00| 0:0:04| chol 1 1
35|0.754|1.000|1.2e-14|8.2e-13|7.8e-05|-1.788065e+00 -1.788126e+00| 0:0:04| chol 1 1
36|0.741|0.901|8.4e-15|7.6e-13|4.1e-05|-1.788083e+00 -1.788114e+00| 0:0:04| chol 1 1
37|0.780|0.869|7.8e-15|4.2e-13|1.9e-05|-1.788093e+00 -1.788108e+00| 0:0:04| chol 1 1
38|0.773|0.861|1.1e-14|2.1e-13|9.0e-06|-1.788097e+00 -1.788104e+00| 0:0:04| chol 1 1
39|0.784|0.848|1.0e-14|1.0e-13|4.1e-06|-1.788099e+00 -1.788102e+00| 0:0:04| chol 1 1
40|0.776|0.842|1.3e-14|4.7e-14|1.9e-06|-1.788100e+00 -1.788102e+00| 0:0:04| chol 1 1
41|0.779|0.837|2.2e-14|2.2e-14|8.8e-07|-1.788101e+00 -1.788101e+00| 0:0:04| chol
linsysolve: Schur complement matrix not positive definite
switch to LU factor. lu 1 1
42|0.774|0.835|2.5e-14|1.0e-14|4.1e-07|-1.788101e+00 -1.788101e+00| 0:0:04| lu 1 1
43|0.775|0.832|1.9e-14|4.9e-15|1.9e-07|-1.788101e+00 -1.788101e+00| 0:0:04| lu 1 1
44|0.773|0.830|1.7e-14|2.3e-15|9.0e-08|-1.788101e+00 -1.788101e+00| 0:0:04| lu 2 1
45|0.774|0.828|9.2e-15|1.1e-15|4.2e-08|-1.788101e+00 -1.788101e+00| 0:0:04|
stop: max(relative gap, infeasibilities) < 1.49e-08

number of iterations = 45
primal objective value = -1.78810090e+00
dual objective value = -1.78810093e+00
gap := trace(XZ) = 4.23e-08
relative gap = 9.24e-09
actual relative gap = 6.99e-09
rel. primal infeas (scaled problem) = 9.22e-15
rel. dual " " " = 1.10e-15
rel. primal infeas (unscaled problem) = 0.00e+00
rel. dual " " " = 0.00e+00
norm(X), norm(y), norm(Z) = 1.6e+06, 2.1e+15, 2.1e+15
norm(A), norm(b), norm(C) = 1.9e+01, 1.0e+00, 1.2e+01
Total CPU time (secs) = 4.36
CPU time per iteration = 0.10
termination code = 0
DIMACS: 9.2e-15 0.0e+00 6.8e-15 0.0e+00 7.0e-09 9.2e-09

Status: Solved
Optimal value (cvx_optval): +1.7881

3

Thanks for your reply. I further want to know why the results of the three solvers are different.
Which solver should I use?
In my opinion, a robust code should be solvable with every solver .

4

SDPT3 did the best. Both SDPT3 and SeDuMi produced optimal W and R0 which are rank one pdf, with largest eigenvalue of 0.5 (within roundoff). They also both satisfied the trace constraint, but SDPT3 did so with equality, whereas SeDuMi did not. Perhaps that’s why SDPT3 got a little better objective. Clearly, the problem is actually (primal) feasible.

However, aside from the small magnitude input data which might be causing difficulties for the solvers, you have also provided redundant semidefinite constraints by virtue of semidefinite declaration and the redundant >= 0 constraints, which in sdp mode, constrain the LHS to be semidefinite. Removing the redundant >= 0 constraints results in the solvee being provided different input, even though from a mathematical perspective, the problems are equivalent. SDPT3 does about the same, but SeDuMI produces a very different, but still feasible solution, and with a better objective.

Mosek produced this result with the redundant >= 0 constraints removed: Perhaps this would actually be a usable solution, but Mosek is overfussy, and CVX is overly conservative in converting Mosek’s status into failed and refusing to provide the results to the user.

Calling Mosek 10.1.10: 321 variables, 129 equality constraints

MOSEK Version 10.1.12 (Build date: 2023-9-20 10:51:32)
Copyright (c) MOSEK ApS, Denmark WWW: mosek.com
Platform: Windows/64-X86

Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 129
Affine conic cons. : 0
Disjunctive cons. : 0
Cones : 64
Scalar variables : 193
Matrix variables : 2 (scalarized: 272)
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. - primal attempts : 1 successes : 1
Lin. dep. - dual attempts : 0 successes : 0
Lin. dep. - primal deps. : 0 dual deps. : 0
Presolve terminated. Time: 0.00
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 65
Optimizer - Cones : 8
Optimizer - Scalar variables : 81 conic : 24
Optimizer - Semi-definite variables: 2 scalarized : 272
Factor - setup time : 0.00
Factor - dense det. time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 2145 after factor : 2145
Factor - dense dim. : 0 flops : 2.69e+06
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 3.1e+01 1.0e+00 1.0e+00 0.00e+00 8.000000000e-07 0.000000000e+00 1.0e+00 0.02
1 3.1e+00 1.0e-01 1.4e-01 -5.53e-01 3.336450191e-06 1.484640831e+00 1.0e-01 0.02
2 4.9e-01 1.6e-02 1.3e-02 1.93e-01 9.170503821e-06 6.167983691e-01 1.6e-02 0.02
3 1.2e-01 3.9e-03 1.9e-03 7.93e-01 2.941257947e-05 1.996000337e-01 3.9e-03 0.03
4 2.5e-02 8.1e-04 3.0e-04 4.93e-01 8.327717903e-05 1.273892580e-01 8.1e-04 0.03
5 5.4e-03 1.7e-04 5.9e-05 2.09e-01 2.031294206e-04 1.100303363e-01 1.7e-04 0.03
6 1.2e-03 3.8e-05 1.2e-05 1.80e-01 4.700684824e-04 1.050365880e-01 3.8e-05 0.03
7 2.4e-04 7.8e-06 2.4e-06 1.77e-01 1.059065457e-03 9.240383398e-02 7.8e-06 0.05
8 4.6e-05 1.5e-06 4.6e-07 -2.11e-02 2.348920119e-03 9.581473196e-02 1.5e-06 0.05
9 1.1e-05 3.5e-07 1.0e-07 -3.28e-02 4.995314466e-03 9.346184630e-02 3.5e-07 0.05
10 2.2e-06 5.7e-07 2.2e-08 -1.05e-01 1.102340318e-02 1.071276252e-01 6.9e-08 0.05
11 4.9e-07 1.7e-06 4.8e-09 5.45e-02 2.393249338e-02 1.162049794e-01 1.6e-08 0.06
12 9.7e-08 6.7e-06 9.2e-10 7.32e-02 5.316191086e-02 1.396575119e-01 3.1e-09 0.06
13 1.9e-08 3.8e-05 1.8e-10 -8.72e-03 1.151059796e-01 1.997578674e-01 6.2e-10 0.06
14 4.2e-09 3.2e-04 3.8e-11 -4.38e-02 2.428442279e-01 3.239930097e-01 1.3e-10 0.06
15 1.0e-09 8.5e-04 8.9e-12 3.34e-02 4.833281814e-01 5.565332848e-01 3.3e-11 0.08
16 2.2e-10 4.1e-03 1.6e-12 1.56e-01 9.303090676e-01 9.833775730e-01 7.1e-12 0.08
17 3.9e-11 8.5e-03 1.9e-13 2.90e-01 1.566247035e+00 1.590574670e+00 1.2e-12 0.08
18 5.0e-12 2.3e-02 6.3e-15 8.07e-01 1.867688287e+00 1.869335892e+00 1.6e-13 0.08
19 9.7e-14 3.2e-02 3.3e-17 1.16e+00 1.788772911e+00 1.788849330e+00 4.6e-15 0.08
20 1.3e-13 1.9e-03 1.0e-16 1.01e+00 1.788166020e+00 1.788170058e+00 2.4e-16 0.09
21 1.7e-13 2.6e-03 2.5e-17 1.00e+00 1.788335304e+00 1.788337550e+00 1.3e-16 0.09
22 3.0e-13 3.7e-03 1.2e-16 1.00e+00 1.788382541e+00 1.788383760e+00 7.3e-17 0.11
23 3.0e-13 3.6e-03 1.5e-17 9.99e-01 1.788383257e+00 1.788384484e+00 7.3e-17 0.11
24 3.0e-13 4.8e-03 4.0e-17 1.00e+00 1.788460569e+00 1.788461229e+00 3.9e-17 0.12
25 2.6e-13 4.8e-03 5.1e-19 8.84e-01 1.788460571e+00 1.788461224e+00 3.9e-17 0.12
26 2.6e-13 4.8e-03 2.7e-17 9.99e-01 1.788460466e+00 1.788461124e+00 3.9e-17 0.14
27 2.9e-13 4.8e-03 9.5e-17 9.79e-01 1.788460357e+00 1.788461026e+00 3.9e-17 0.14
28 3.3e-13 4.8e-03 1.2e-16 1.01e+00 1.788460334e+00 1.788461008e+00 3.9e-17 0.16
29 3.0e-13 3.5e-03 1.8e-16 1.00e+00 1.788365498e+00 1.788366030e+00 3.0e-17 0.16
30 3.3e-13 3.5e-03 1.0e-17 1.00e+00 1.788365383e+00 1.788365879e+00 3.0e-17 0.17
31 2.9e-13 3.5e-03 6.6e-17 9.99e-01 1.788365272e+00 1.788365783e+00 3.0e-17 0.17
32 2.9e-13 3.5e-03 6.6e-17 9.99e-01 1.788365272e+00 1.788365783e+00 3.0e-17 0.19
33 3.0e-13 3.5e-03 1.1e-16 1.00e+00 1.788365253e+00 1.788365772e+00 3.0e-17 0.20
34 3.0e-13 3.5e-03 1.1e-16 1.00e+00 1.788365253e+00 1.788365772e+00 3.0e-17 0.20
35 3.0e-13 3.5e-03 1.1e-16 1.00e+00 1.788365253e+00 1.788365772e+00 3.0e-17 0.22
Optimizer terminated. Time: 0.23

Interior-point solution summary
Problem status : ILL_POSED
Solution status : DUAL_ILLPOSED_CER
Primal. obj: 1.3391179395e-05 nrm: 1e+01 Viol. con: 7e-06 var: 0e+00 barvar: 0e+00 cones: 0e+00
Optimizer summary
Optimizer - time: 0.23
Interior-point - iterations : 36 time: 0.23
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: Failed
Optimal value (cvx_optval): NaN

I ran this in YALMIP with Mosek 10.1.12 as solver, which resulted in Status UNKNOWN, the solution is primal feasible, but the optimal objective vale is much worse then produced by SDPT3 or SeDuMi in CVX. Those solvers have numerical problems and are all over the map in YALMIP. So my conclusion is that the combination of problem formulation and input data is not very good.

5

Another contributor to difficulty might be that only the sum W + R0 is well-determined, not the individual matrices. So you could eliminate R0 from the formulation.