Wednesday, November 20, 2019

MIP solver stopping criteria

For larger MIP models we often don't wait for a proven optimal solution. This just takes too long, and actually we are spending a lot of time in proving optimality without much return in terms of better solutions. There are a number of stopping criteria that are typically available:

  1. Time Limit : stop after \(x\) seconds (or hours)
  2. Relative Gap: stop if gap between best possible bound and best found integer solution becomes less than \(x\%\).  Different solvers use different definitions (especially regarding the denominator).
  3. Absolute Gap: similar to relative gap, but can be used when the relative gap cannot be computed (division by zero or small number).
  4. Node Limit: stop on number of explored branch & bound nodes.
  5. Iteration Limit: stop on number of Simplex iterations. This number can be huge.

I have ordered these stopping criteria in how useful they are (to me). Time limit is by far the most important: just tell the solver how long we are willing to wait. Stopping on an achieved gap is also useful. I don't remember ever using a node or iteration limit.

If you specify several limits, typically a solver will stop as soon as it hits any one of the specified limits. In other words: multiple stopping criteria are combined in an "or" fashion.

When stopping on a time limit, it is still important to inspect the final gap. A small gap gives us a guaranteed quality assurance about the solution.

For large models the tail is often very long, and we probably see hardly any movement: no new integer solutions are found and the best bound is moving very slowly (and moving less over time). So I really want to stop if there is not much hope for a better solution.

I would suggest another possible stopping criterion:

stop if the time since the last new (and improving) integer solution exceeds a time limit

If the time since that last new integer solution is large, we can infer that the probability of finding a better solution is small. We can also interpret this as resetting the clock after each new integer solution. I don't think any solver has this. Of course, for some solvers, we can implement this ourselves using some callback function.

1 comment:

  1. Totally agree - I have done this using e.g. callbacks for a number of projects.

    I would add another refinement that I have used and that is to keep track of the difference in the objective value between the current incumbent and what it was some fixed time ago. Then we can stop if the solver has not made at least 0.1% improvement in the objective value during the last 10 minutes for example. This is useful if there are very many similar solutions near to the supposed optimal value, and we might get a prolonged sequence of solutions that are improving but only by a tiny amount.

    To paraphrase: as long as you are making *worthwhile* progress in finding better solutions, keep going.