solver return infeasible for a solvable problem

[zying0827]

Hello,

When I input the attached ISDP problem, the solver returned "infeasible". However, there are some feasible solutions to this problem, such as x_2, x_5, and x_11 are 1, and 0 for the rest variables. What's more interesting is that when I add the 53th constraint x_5>=1, the solver returns "optimal solution found", which is unreasonable since adding constraints will not enlarge the solution space. Could you please provide some guidance? Thanks in advance.

The following is the content of test.dat-s, which is a binary SDP problem.

15
2
4 -52
0 1 2 0 1 2 0 0 0 0 0 0 0 0 0 
0 1 1 1 -17834.694217631153
0 1 1 2 165.3057823691147
0 1 1 3 -291.61806365614996
0 1 1 4 -1.0
0 1 2 2 -1834.694217630329
0 1 2 3 -291.61806365605264
0 1 2 4 -2.0
0 1 3 3 -52.08506915514171
0 1 3 4 -1.0
0 1 4 4 -4.0
1 1 1 1 160006.66555574554
1 1 1 2 -1664.7225462411939
1 1 1 3 416.5972337945695
2 1 1 1 -12000.666555574515
2 1 1 2 332.9445092482388
2 1 1 3 -83.3194467589139
3 1 1 1 18667.555407432712
3 1 1 2 -332.9445092482388
3 1 1 3 83.3194467589139
4 1 1 2 -13329.445092483375
4 1 2 2 46685.5524079284
4 1 2 3 833.1944675886523
5 1 1 2 2665.889018496675
5 1 2 2 -1334.4442592900086
5 1 2 3 -166.63889351773045
6 1 1 2 -2665.889018496675
6 1 2 2 4001.9996667218993
6 1 2 3 166.63889351773045
7 1 1 2 -33327.77870354647
8 1 1 2 6665.555740709295
9 1 1 2 -6665.555740709295
10 1 1 2 6665.555740709295
11 1 1 2 -1333.1111481418588
12 1 1 2 1333.1111481418588
13 1 1 2 -6665.555740709295
14 1 1 2 1333.1111481418588
15 1 1 2 -1333.1111481418588
1 2 1 1 1
1 2 2 2 -1
0 2 2 2 -1
2 2 3 3 1
2 2 4 4 -1
0 2 4 4 -1
3 2 5 5 1
3 2 6 6 -1
0 2 6 6 -1
4 2 7 7 1
4 2 8 8 -1
0 2 8 8 -1
5 2 9 9 1
5 2 10 10 -1
0 2 10 10 -1
6 2 11 11 1
6 2 12 12 -1
0 2 12 12 -1
7 2 13 13 1
1 2 14 14 1
7 2 14 14 -1
4 2 15 15 1
7 2 15 15 -1
7 2 16 16 1
1 2 16 16 -1
4 2 16 16 -1
0 2 16 16 -1
8 2 17 17 1
1 2 18 18 1
8 2 18 18 -1
5 2 19 19 1
8 2 19 19 -1
8 2 20 20 1
1 2 20 20 -1
5 2 20 20 -1
0 2 20 20 -1
9 2 21 21 1
1 2 22 22 1
9 2 22 22 -1
6 2 23 23 1
9 2 23 23 -1
9 2 24 24 1
1 2 24 24 -1
6 2 24 24 -1
0 2 24 24 -1
10 2 25 25 1
2 2 26 26 1
10 2 26 26 -1
4 2 27 27 1
10 2 27 27 -1
10 2 28 28 1
2 2 28 28 -1
4 2 28 28 -1
0 2 28 28 -1
11 2 29 29 1
2 2 30 30 1
11 2 30 30 -1
5 2 31 31 1
11 2 31 31 -1
11 2 32 32 1
2 2 32 32 -1
5 2 32 32 -1
0 2 32 32 -1
12 2 33 33 1
2 2 34 34 1
12 2 34 34 -1
6 2 35 35 1
12 2 35 35 -1
12 2 36 36 1
2 2 36 36 -1
6 2 36 36 -1
0 2 36 36 -1
13 2 37 37 1
3 2 38 38 1
13 2 38 38 -1
4 2 39 39 1
13 2 39 39 -1
13 2 40 40 1
3 2 40 40 -1
4 2 40 40 -1
0 2 40 40 -1
14 2 41 41 1
3 2 42 42 1
14 2 42 42 -1
5 2 43 43 1
14 2 43 43 -1
14 2 44 44 1
3 2 44 44 -1
5 2 44 44 -1
0 2 44 44 -1
15 2 45 45 1
3 2 46 46 1
15 2 46 46 -1
6 2 47 47 1
15 2 47 47 -1
15 2 48 48 1
3 2 48 48 -1
6 2 48 48 -1
0 2 48 48 -1
1 2 49 49 1
2 2 49 49 1
3 2 49 49 1
0 2 49 49 1
1 2 50 50 -1
2 2 50 50 -1
3 2 50 50 -1
0 2 50 50 -1
4 2 51 51 1
5 2 51 51 1
6 2 51 51 1
0 2 51 51 1
4 2 52 52 -1
5 2 52 52 -1
6 2 52 52 -1
0 2 52 52 -1
*This is constraint (x_5 >= 1)
*5 2 53 53 1
*0 2 53 53 1
*INTEGER
*1
*2
*3
*4
*5
*6
*7
*8
*9
*10
*11
*12
*13
*14
*15

[pfetsch]

When I use Mosek as SDP-solver, the solution of the first SDP fails with status "UNKNOWN". SCIP-SDP then tries to solve a penalty formulation, for which Mosek says that one cannot drive the penalty term to 0. But it is only very slightly off, with a value of 1.45e-05 (the tolerance is 1e-5). Thus, SCIP-SDP decides that the relaxation and thus the problem is infeasible.

When I use DSDP, the problem is solved without problem. The same happens when I only solve LPs.

I am not sure why Mosek cannot solve the relaxation. However, the matrices are "not nice" in the sense that they contain values of very different magnitude (160006.6 vs. -1).