Re: The coefficient matrix is not full row rank, numerical problems may occur. sedumi 1.3 is it a problem?

[Johan Löfberg]

solved without proble,ms here using mosek. However, your problem is not well posed as 0 appears to be feasible

>> optimize(Constraints)

Problem

  Name                   :                 

  Objective sense        : min             

  Type                   : CONIC (conic optimization problem)

  Constraints            : 21              

  Cones                  : 0               

  Scalar variables       : 3               

  Matrix variables       : 38              

  Integer variables      : 0               

Optimizer started.

Presolve started.

Linear dependency checker started.

Linear dependency checker terminated.

Eliminator started.

Freed constraints in eliminator : 1

Eliminator terminated.

Eliminator - tries                  : 1                 time                   : 0.00            

Lin. dep.  - tries                  : 1                 time                   : 0.01            

Lin. dep.  - number                 : 0               

Presolve terminated. Time: 0.03    

Problem

  Name                   :                 

  Objective sense        : min             

  Type                   : CONIC (conic optimization problem)

  Constraints            : 21              

  Cones                  : 0               

  Scalar variables       : 3               

  Matrix variables       : 38              

  Integer variables      : 0               

Optimizer  - threads                : 2               

Optimizer  - solved problem         : the primal      

Optimizer  - Constraints            : 20

Optimizer  - Cones                  : 1

Optimizer  - Scalar variables       : 3                 conic                  : 3               

Optimizer  - Semi-definite variables: 38                scalarized             : 228             

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 : 210               after factor           : 210             

Factor     - dense dim.             : 0                 flops                  : 3.77e+04        

ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  

0   1.6e+01  1.0e+00  1.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   1.0e+00  0.06  

1   5.0e+00  3.2e-01  5.6e-01  1.00e+00   0.000000000e+00   0.000000000e+00   3.2e-01  0.17  

2   1.7e+00  1.1e-01  3.3e-01  1.00e+00   0.000000000e+00   0.000000000e+00   1.1e-01  0.17  

3   8.3e-01  5.3e-02  2.3e-01  1.00e+00   0.000000000e+00   0.000000000e+00   5.3e-02  0.19  

4   1.9e-01  1.2e-02  1.1e-01  1.00e+00   0.000000000e+00   0.000000000e+00   1.2e-02  0.19  

5   3.5e-03  2.2e-04  1.5e-02  1.00e+00   0.000000000e+00   0.000000000e+00   2.2e-04  0.19  

6   7.6e-09  4.8e-10  4.8e-10  1.00e+00   0.000000000e+00   0.000000000e+00   4.8e-10  0.23  

Optimizer terminated. Time: 0.28    

Interior-point solution summary

  Problem status  : PRIMAL_AND_DUAL_FEASIBLE

  Solution status : OPTIMAL

  Primal.  obj: 0.0000000000e+00    nrm: 1e+01    Viol.  con: 2e-03    var: 0e+00    barvar: 0e+00  

  Dual.    obj: 0.0000000000e+00    nrm: 9e-09    Viol.  con: 0e+00    var: 3e-10    barvar: 5e-10  

Optimizer summary

  Optimizer                 -                        time: 0.28    

    Interior-point          - iterations : 6         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    

ans = 

  struct with fields:

    yalmiptime: 0.2545

    solvertime: 0.4025

          info: 'Successfully solved (MOSEK)'

       problem: 0

[elmajidi azeddine]

Hello Johan,

Thanks for your quick answer, i'll try to install mosek and rerun the code, however u said that my problem is not well posed, 

has it a relation with my strict inequalities in the theorem that i ommited in the code (for Q and the constraints with the increment i)?

there is it a way to overcome this (to be well posed)?

why sedumi can't resolve the problem?

and sorry again if i ask stupid question.

Thks in advance

[Johan Löfberg]

Strict inequalities are impossible in practice (as there is no minimizer). You have to dehomogenize it, by adding some explicit bound (such as replacing P>0 with P>=eye(n)) .

Sedumi simply doesn't seem to like you problem, likely due to the homogeneuous nature

[Johan Löfberg]

Ading Q>=eye(3) to dehomogenize leads to infeasibility, so I would suspect you problem is not strictly feasible

[elmajidi azeddine]

Hello again,

I tried ur instruction by using mosek instead of sedumi to resolve the problem without changing anything in my constraint, and as u said the problem got a solution that magically and hoepfully stabilized the system in my previous stacked point!!! Thanks a lot.

However i have final question to you, if you remember my LMIs were like:

• Q>0 
  
• S>=0
  
• Q*Ai'+Ai*Q+C'*Ni'*Bi'+Bi*Ni*C+S<0 for each i
  
• Q*(Ai'+Aj')+(Ai+Aj)*Q+C'*(Ni'*Bj'+Nj'*Bi')+(Bi*Nj+Bj*Ni)*C-2*S<=0 for each i <j
  
• C*Q=M*C

and to verifiy the compliance of my solution i always try to see the nature of the gotten eigenvalues for S, Q, ...

in latest try with mosek I got eig(Q)>0 same thing with eig(S), but for Q*Ai'+Ai*Q+C'*Ni'*Bi'+Bi*Ni*C+S when I try eig(Qfeasible*A(1:3,1:3)'+A(1:3,1:3)*Qfeasible+B(1:3,1)*Nfeasible(1)*C+C'*Nfeasible(1)'*B(1:3,1)'+Sfeasible) for i=1 I got strictly positive values instead of negative

Is it normal or did i something wrong in my procedure ?

Sorry if I took a lot of your time

[Johan Löfberg]

As I said, your problem is likely infeasible. When you have a homogeneous problem, the solver simply returns 0 or something around that which is close enough, but of course garbage if you must have strictly feasible solution

The problem to find x>0 and x<0 is trivially infeasible, but a numerical solver might return x = 0 and think it is good enough as it violates feasibility very little

[Johan Löfberg]

btw, cells would simplify your coding a lot. Use  A = {A1,A2,A3,..} etc instead, and you can simply use A{i}

[elmajidi azeddine]

Thanks a lot for your directive Johan