Mosek Feasibility with negative Primal Residuals

[ers...@gmail.com]

Hello!
I am a Yalmip user and I have few questions;

1- In order to explain my question I am gonna share a part of my code and corresponding output.

 diagnostics = optimize([LMI,gamma<=15],[],ops);

The output: 

Mosek error: MSK_RES_TRM_STALL ()

 

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

|   ID|               Constraint|   Primal residual|   Dual residual|              Tag|

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

|   #1|        Matrix inequality|          -0.41977|      2.9993e-13|           LMI 32|

|   #2|        Matrix inequality|       -4.1308e-09|      5.6593e-13|           LMI 14|

|   #3|        Matrix inequality|           -2.9014|      2.8338e-13|           LMI 36|

|   #4|        Matrix inequality|          1.78e-08|      9.0485e-09|           LMI Q1|

|   #5|        Matrix inequality|       -8.7006e-14|           1.987|           LMI Q2|

|   #6|   Elementwise inequality|        0.00066351|      1.4255e-09|   Positive Gamma|

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

 

diagnostics.problem=-1 

When I set the gamma is less than 15 I could not achieve any result. On the other hand, when the upper bound of gamma is set to 8 which makes LMIs more conservative, Mosek can find a feasible solution;

 diagnostics = optimize([LMI,gamma<=8],[],ops);

The output: 

Mosek error: MSK_RES_TRM_STALL ()

 

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

|   ID|               Constraint|   Primal residual|   Dual residual|              Tag|

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

|   #1|        Matrix inequality|        7.8105e-07|      8.4085e-09|           LMI 32|

|   #2|        Matrix inequality|        7.2851e-05|      2.1125e-08|           LMI 14|

|   #3|        Matrix inequality|       -1.9863e-08|      7.1241e-09|           LMI 36|

|   #4|        Matrix inequality|        0.00095311|      6.2623e-05|           LMI Q1|

|   #5|        Matrix inequality|       -9.0694e-09|      18378.9679|           LMI Q2|

|   #6|   Elementwise inequality|            7.4971|      3.1213e-05|   Positive Gamma|

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

 

diagnostics.problem=0

gamma =

    7.4971

I cannot make any comments on these results. The relaxed conditions cannot find consider system feasible despite the fact that more conservative one conclude feasibility. My expected gamma is around 10. According to results both of them contains negative primal residuals. What is the origin of this dilemma? Yes I it possible to formalize the problem as non-convex 

but both of them starts with

Problem

  Name                   :                 

  Objective sense        : min             

  Type                   : CONIC (conic optimization problem)

Lines. Or We cannot trust the solution when the Mosek error: MSK_RES_TRM_STALL () is displayed? 

My second question is that;Since the solver produces negative primal residuals, how mosek determines feasibility? Is it related to tolerance of solver? If so what is the tolerance? 

Thank  you so much!

Best Regards,
Muhammed Ersat EMEK

[Erling D. Andersen]

By default MOSEK prints out a solution summary including relevant information about the solution. See

   http://docs.mosek.com/8.0/toolbox/tutorial-solution-analysis.html

Please include the solution summary in a reply if you want us to comment.

Regard how the MOSEK optimizer works then you can get an idea by reading

   https://scholar.google.dk/citations?view_op=view_citation&hl=da&user=3I3Gs2cAAAAJ&citation_for_view=3I3Gs2cAAAAJ:9yKSN-GCB0IC

 

[ers...@gmail.com]

Thanks Erling,

I read the first link but the second one is broken. In my case there is not any optimization, I am only checking the feasibility and I was using check(constraints) command for primal and dual residuals test. Then according to your direction I'll change my interpretation. Instead of check command you said the Problem and Solution Status of Mosek is more reliable even primal and dual residual output of check command are negative.

For first case which gamma<=15 

  Problem status  : ILL_POSED

  Solution status : DUAL_ILLPOSED_CER

The first one should be feasible but its ill posed

For  second case gamma<=9

  Problem status  : ILL_POSED

  Solution status : DUAL_ILLPOSED_CER

the second one should be feasible, it is not infeasible but ill posed

For  second case gamma<=8

  Problem status  : PRIMAL_AND_DUAL_FEASIBLE

  Solution status : NEAR_OPTIMAL

the thid one have to be infeasible but it is dual feasible.

How this can happen? 

[Erling D. Andersen]

Given MOSEK says ILL POSED then your problem seems to be nasty problem. And hence MOSEK has problems computing an accurate solution. 

So the right question is: Is your model a good well posed model?

You know garbage in garbage out. Sorry, but that is the way it is,