Interpretation of residuals returned by "check"

[Pan Zhao]

Hi Johan,

I am trying to use command "check" to check feasibility of all the constraints in my optimization problem. For this, I would like to understand whether negative residuals mean infeasibility. 

In the help doc for "check" (http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Commands.Check), it is first stated that "a solution is feasible if all residuals related to inequalities are non-negative". So non-negative residuals mean feasibility. However, later, another sentence says "Also note that check only computes residuals. It does not judge if these indicate infeasibility". I am a little confused because here it says residuals cannot be used to judge feasibility. 

Any clarification will be appreciated. 

Thanks

Pan

[Johan Löfberg]

It means that you have to make a practical judgment from the actual application as to whether the tiny negative residuals means that the solution is practically infeasible. If your optimization problem is used to build a road intended to be at least 10 meters wide, and the solution is 9.99999999999 meters, do you care? 

[Pan Zhao]

Thank you Johan for your reply. I am clear now.  So the conclusion is: non-negative residuals indicate feasibility; however, negative residuals do not indicate infeasibility. 

[Johan Löfberg]

No Non-negative means feasible, negative means feasible.

What I say is that a measure of feasibility has to be judged w.r.t the context. If your constraint is that you should build a 10 meter wide road and the solution turns out to be 9.9999999, it is an infeasible solution, but who cares. For all practical purposes it is feasible. However, if your task is to prove stability of a very important dangerous system, and stability is guaranteed if a variable x is non-negative, and it turns out to be 0.000000001, i.e., feasible, would you bet your life that it still works?

[Johan Löfberg]

meant Non-negative means feasible, negative means infeasible

[Pan Zhao]

Thanks for your further clarification. I think I got the idea. Non-negative means feasible, negative means infeasible. However, infeasible solution can be regarded as feasible if the (absolute value of ) negative residuals are very small. 

[Johan Löfberg]

..depending on the context, it's up to you to make an engineering judgement on the solution