Problem is feasible without an objective function, infeasible with (objective is bounded)

[Tom Holden]

Hi,

The attached example "ProblemTest.m" illustrates a situation in which when run with an objective, the solver (Mosek or Gurobi) reports it is infeasible, but when run without it reports a feasible solution.

To run the file, you just need to have the "ProblemWorkspace.mat" file in the same folder you are running from.

Note also that the objective is bounded below, as the objective is given by:

  Objective = -( sum( Switch ) * numel( Buffer ) * 10 + sum( Buffer(:) ) );

where Switch is a vector with elements equal to 1, 0 or -1, and Buffer is a vector with elements all less than 0.5.

Do you have any idea what is causing this? I expect it is to do with the fact that Switch is defined as:

  Switch = sign( Buffer - 1e-4 );

Do you have any suggestions for a work-around? I have solved many similar problems without issue.

Thanks in advance,

Tom

[Johan Löfberg]

YALMIP assumes models are well-scaled when doing big-M modelling, so unbounded variables are capped at 10^4. This is not feasible in your model which appears to be badly scaled (solving the feasibility problem leads to buffer in the order of 10^6)

[Tom Holden]

Is there an option to change that value? Is it possible to find out why YALMIP is resorting to Big-M bounds? It shouldn't be necessary here I thought.

[Johan Löfberg]

Not unless you hack the code.

sign is an operator which requires a big-M model (and it is a horrible operator to model as it is discontinuous, even worse that it has a point with a specific value). 

 Preferred alternative is to model the sign operator manually, to understand why it is such a bad operator to work with, and how that can lead to very bad results

[Tom Holden]

Do you have any suggestions for this particular example? Essentially I want as many elements of Buffer as possible to be positive. Conditional on that, I want the elements of Buffer to be as large as possible.

[Johan Löfberg]

Use other units (don't work with euro, work with MEUR etc)

But for your practical problem here, don't model that with sign. Best is always to understand what the MILP model sohlud be, in this case maximize sum(d) where d is a binary vector and Buffer >= (1-d)*lowerbound

[Tom Holden]

Thanks for the advice!