how to solve a SDP problem or LMI with compelx data?

[Ricardo May]

in addition, W contains complex numbers, and W is expected to be hermitian symmetric.

what I want to solve is just to recover a theorem in a paper I read, which was solved by SEDUMI. However, something occurred in my codes. to do that, I have write my codes just using SEDUMI and YALMIP, but I am not familiar with the K structure in SEDUMI, which may cause some mistakes. in addition, when I run my codes in MATLAB, here comes some unacceptable problems.

if you have any advice, please help me.

[Johan Löfberg]

Your YALMIP code doesn't make sense as you defined a complex-valued objective

>> obj

Linear scalar (complex, 22 variables)

[Ricardo May]

sorry, I have attached a wrong file, here is the exact one I am proceeding. but after I update the latest YALMIP, a new problem occurred.

MOSEK Version 7.1.0.12 (Build date: 2014-12-12 11:23:25)

Copyright (c) 1998-2014 MOSEK ApS, Denmark. WWW: http://mosek.com

Platform: Windows/64-X86

Computer

  Platform               : Windows/64-X86  

  Cores                  : 2               

Problem

  Name                   :                 

  Objective sense        : min             

  Type                   : CONIC (conic optimization problem)

  Constraints            : 81              

  Cones                  : 0               

  Scalar variables       : 1               

  Matrix variables       : 400             

  Integer variables      : 0               

Optimizer started.

Optimizer terminated. Time: 0.03    

Optimizer summary

  Optimizer                 -                        time: 0.03    

    Interior-point          - iterations : 0         time: 0.00    

      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    

        Clean primal-dual   - iterations : 0         time: 0.00    

    Simplex                 -                        time: 0.00    

      Primal simplex        - iterations : 0         time: 0.00    

      Dual simplex          - iterations : 0         time: 0.00    

      Primal-dual simplex   - iterations : 0         time: 0.00    

    Mixed integer           - relaxations: 0         time: 0.00    

Mosek error: MSK_RES_ERR_MISSING_LICENSE_FILE (MOSEK cannot locate a license. The default search path is 'C:\Users\Ricardo\mosek\mosek.lic'.)

ans = 

    yalmiptime: NaN

    solvertime: NaN

          info: 'Unknown problem in solver (Turn on '...'

       problem: 9

ans =

   NaN

by the way, could please give me some instructions about the K structure in SEDUMI, it is hard for me to understand K structure using the Guide, thank you!

[Johan Löfberg]

Well, now you've installed mosek for which you obviously have no license visible. Select sedumi as solver through sdpsettings if you want to use it

[Ricardo May]

SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.

Alg = 2: xz-corrector, theta = 0.250, beta = 0.500

eqs m = 81, order n = 2403, dim = 14403, blocks = 402

nnz(A) = 93609 + 0, nnz(ADA) = 6561, nnz(L) = 3321

 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec

  0 :            1.07E+02 0.000

  1 :  -6.79E-02 6.79E+01 0.000 0.6363 0.9000 0.9000   5.75  1  1  1.3E+02

  2 :  -7.90E-02 2.12E+01 0.000 0.3125 0.9000 0.9000   5.73  1  1  1.4E+01

  3 :   8.09E-03 1.66E+00 0.000 0.0781 0.9900 0.9900   0.88  1  1  1.7E+00

  4 :   1.64E+00 1.60E-01 0.000 0.0962 0.9900 0.9900  -1.11  1  1  5.5E+00

  5 :   2.16E+01 1.15E-02 0.000 0.0723 0.9900 0.9900  -1.13  1  1  5.5E+00

  6 :   3.26E+02 7.64E-04 0.000 0.0662 0.9900 0.9900  -1.01  1  1  5.5E+00

  7 :   4.98E+03 5.02E-05 0.000 0.0657 0.9900 0.9900  -1.00  1  1  5.5E+00

  8 :   7.61E+04 3.29E-06 0.000 0.0655 0.9900 0.9900  -1.00  3  2  5.5E+00

  9 :   1.22E+06 2.00E-07 0.000 0.0608 0.9900 0.9900  -1.00  5  7  5.4E+00

 10 :   1.67E+07 1.44E-08 0.000 0.0722 0.9900 0.9900  -1.00  9 17  5.4E+00

 11 :   2.58E+08 9.37E-10 0.000 0.0649 0.9900 0.9900  -1.00 19 21  5.4E+00

Run into numerical problems.

Primal infeasible, dual improving direction found.

iter seconds  |Ax|    [Ay]_+     |x|       |y|

 11     19.9   1.8e-08   2.7e-09   4.5e+00   1.3e+01

Detailed timing (sec)

   Pre          IPM          Post

1.560E-01    1.260E+01    9.800E-02    

Max-norms: ||b||=2.805704e-01, ||c|| = 1,

Cholesky |add|=0, |skip| = 8, ||L.L|| = 1.06693.

ans = 

    yalmiptime: 1.9756

    solvertime: 12.8994

          info: 'Unbounded objective function (SeDuMi...'

       problem: 2

ans =

    -1

but I finally got an unbounded objective funtion with the latter SDP_YALMIP.m

[Johan Löfberg]

Yes, the model you've created is unbounded. Simply look at the solutions you get when you artificially constrain it. Now take that solution and understand which constraint you've forgot, or declared incorrectly

optimize([F,norm(W) <= 100],obj) % Increase 100 and it simply generates larger solutions
value(W)