How to identify the effectiveness of results from different solvers

[Zihao Cheng]

The codes in Untitled.m runs different results. I try to solve the Lmis by using mosek and SeDuMi-1.3. And the results confuse me. 

How to explain the results? Which one is the reasonable results that i can believe? 

The simulation displays as followings:

(1) yalmiptime: 2.1260

     solvertime: 76.8530

          info: 'Successfully solved (MOSEK)' 

       problem: 0

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   ID|          Constraint|   Primal residual|   Dual residual|

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   #1|   Matrix inequality|       -3.9494e-12|     -2.1123e-10|

|   #2|   Matrix inequality|       -3.8141e-10|     -6.5572e-10|     (Remark: The negative value of residuals do not illustrate the infeasibility, right?)

|   #3|   Matrix inequality|       -4.1758e-10|     -3.4911e-10|

|   #4|   Matrix inequality|         3086.0946|      7.2239e-11|

|   #5|   Matrix inequality|           38.5738|      5.7942e-13|

|   #6|   Matrix inequality|       -2.8534e-12|     -1.5192e-10|

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

 (2)   yalmiptime: 1.6943

        solvertime: 2.6228e+03

          info: 'Numerical problems (SeDuMi-1.3)'

       problem: 4

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   ID|          Constraint|   Primal residual|   Dual residual|

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

|   #1|   Matrix inequality|         1.142e-09|      8.3986e-13|

|   #2|   Matrix inequality|       -1.5982e-08|      1.0807e-11|

|   #3|   Matrix inequality|       -1.4846e-08|      4.3077e-12|

|   #4|   Matrix inequality|         2357.8364|      1.5187e-09|

|   #5|   Matrix inequality|           39.4067|      1.0856e-11|

|   #6|   Matrix inequality|        1.1722e-09|      8.4009e-13|

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

[Johan Löfberg]

You are solving a problem without objective, so you cannot expect them to return the same solution

You solve with two different solvers, so one might perform better or worse (or use different option to declare numerical issues etc, or not check certain numerical properties)

Negative primal residual means the solution is infeasible. Both your solutions appears to be of normal infeasibility size. No solvers guarantee strictly feasible solutions

[Zihao Cheng]

Thank you very much for your reply.  For your suggestions, i still need to talk with you and respect your answer.

(1) I use "optimize" to solve feasibility problem of LMIs. So, you are right. Feasibility results need not be the same. In other word, would i understand that the same solution can be obtained by different solver if you set the same objective for different solver?

(2)  'https://yalmip.github.io/command/check/ 'says that

 ''Hence, a solution is feasible if all residuals related to inequalities are non-negative. '' (I think that non-negative is the sufficient condition of feasible. And conversely, it does not hold.  Namely, a solution can not be sure of infeasible if there exists some negative residuals  )

''Also note that check only computes residuals. It does not judge if these indicate infeasibility. ''

So, as you said "Negative primal residual means the solution is infeasible." Is it your judgement, but not from the Yalmip. Can you guarantee your judgement is right?

(3) "Both your solutions appears to be of normal infeasibility size. No solvers guarantee strictly feasible solutions" 

What does it means that normal infeasibility size? Can i use the normal infeasibility solutions? 

I still confuse for this normal infeasibility solutions. Does it mean i cheat reader if i put these normal infesibility solution in my paper?

(4) Solver display "successfully solved", but it does not means that the solution is feasibiltiy or not. We must check it after each solving. 

I am so sorry that i have so much troubles.  But these problems are very important in study. Looking forward to your reply.

[Johan Löfberg]

1. No, you can have problems with non-unique optimal solutions.

2,3,4 https://yalmip.github.io/faq/solutionviolated/