[Csdp] Sparse Formulation and Solver

[Duro, Joao]

Hi,

I am currently formulating a SDP problem using for that the C library from CSDP. My problem is described as follows.
I have 9 variables 7 constraints I would like to maximize the trace of C*X and X must be psd.

Objective function coefficients (C in CSDP)
C=[ 2/3 -1/3 -1/3 -1/3 2/3 -1/3 -1/3 -1/3 2/3 ];

Constraint coefficients (Ax in CSDP where x=1..7)
At=[
1 1 1 1 1 1 1 1 1;
1 -1 0 -1 1 0 0 0 0;
1 0 -1 0 0 0 -1 0 1;
0 0 0 0 1 -1 0 -1 1;
1 -1 0 -1 1 0 0 0 0;
0 0 0 0 1 -1 0 -1 1;
1 0 -1 0 0 0 -1 0 1;
];

Constraint equalities (b in CSDP)
b=[0.000000 0.130967 0.231414 0.139552 0.130967 0.139552 0.231414];

Using a solver called Sedumi in Matlab I do:
K.s=[3]; [X]=sedumi(At,b,C,K);
The coefficient matrix is not full row rank, numerical problems may occur.
SeDuMi 1.3 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 = 7, order n = 4, dim = 10, blocks = 2
nnz(A) = 24 + 0, nnz(ADA) = 49, nnz(L) = 28
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            1.67E+01 0.000
  1 :  -1.36E-02 3.19E+00 0.000 0.1905 0.9000 0.9000   1.25  1  1  2.0E+00
  2 :   1.41E-01 5.91E-01 0.000 0.1856 0.9000 0.9000   2.11  1  1  2.6E-01
  3 :   1.67E-01 1.53E-02 0.000 0.0259 0.9900 0.9900   1.39  1  1  5.3E-03
  4 :   1.67E-01 3.59E-07 0.000 0.0000 1.0000 1.0000   1.01  1  1  1.2E-07
  5 :   1.67E-01 3.85E-14 0.000 0.0000 1.0000 1.0000   1.00  1  4  1.3E-14

iter seconds digits       c*x               b*y
  5      0.1  13.7  1.6731100000e-01  1.6731100000e-01
|Ax-b| =   6.1e-15, [Ay-c]_+ =   0.0E+00, |x|=  1.3e-01, |y|=  6.1e-01

Detailed timing (sec)
   Pre          IPM          Post
3.289E-02    1.722E-01    3.330E-02    
Max-norms: ||b||=2.314140e-01, ||c|| = 6.666667e-01,
Cholesky |add|=0, |skip| = 3, ||L.L|| = 2.04072.

The obtained learnt matrix is:
mat(X)

    0.0650   -0.0158   -0.0493
   -0.0158    0.0344   -0.0186
   -0.0493   -0.0186    0.0679


Using CSDP I am getting primal infeasibility as such:

Iter:  0 Ap: 0.00e+00 Pobj: -8.8789832e+00 Ad: 0.00e+00 Dobj:  0.0000000e+00 
Iter:  1 Ap: 8.72e-01 Pobj: -1.7933602e+01 Ad: 1.00e+00 Dobj:  2.0904101e+00 
Iter:  2 Ap: 9.46e-01 Pobj: -4.9484284e-01 Ad: 1.00e+00 Dobj:  2.0335513e+00 
Iter:  3 Ap: 8.00e-01 Pobj: -4.9557359e-02 Ad: 1.00e+00 Dobj:  2.3268098e-01 
Iter:  4 Ap: 1.95e-01 Pobj:  6.4445451e-02 Ad: 3.77e-01 Dobj: -1.7190467e+01 
Iter:  5 Ap: 1.43e-01 Pobj:  1.2187333e-01 Ad: 8.60e-01 Dobj: -4.9203333e+02 
Iter:  6 Ap: 3.07e-03 Pobj:  1.0362899e-01 Ad: 1.19e-01 Dobj: -3.6106653e+04 
Iter:  7 Ap: 8.61e-03 Pobj:  1.3201208e-01 Ad: 6.43e-02 Dobj: -1.8467582e+06 
Iter:  8 Ap: 6.50e-04 Pobj:  1.0846532e-01 Ad: 1.57e-03 Dobj: -5.6943047e+06 
Iter:  9 Ap: 2.93e-03 Pobj:  1.3144031e-01 Ad: 1.00e+00 Dobj: -7.9309967e+09 
Declaring primal infeasibility.
Success: SDP is primal infeasible
Certificate of primal infeasibility: a'*y=-1.00000e+00, ||A'(y)-Z||=2.01564e-10
SDP failed.

The C code for this example is available in the following link http://pastebin.com/sYgbtyHx. 
The generated file with the problem in SDPA sparse format can be found in the following link http://pastebin.com/X4ykk2Sh .. I wonder if there is any mistake considering the values given above.

1st Question: is the problem well formulated in sparse format?

By default Sedumi minimizes (C*X) and CSDP maximizes tr(C*X). 
I have tried to multiply the coefficients (C) with (-1) in both cases, however I still got the same results as mentioned before.

2nd Question: The fact that one solver minimizes and the other maximizes could be an issue?

Let me know if you could spot anything.

Regards,
Joao