Tuning Integer Linear Programming

Change Legacy Options to Improve the Solution Process

Note

Often, you can change the formulation of a MILP to make it more easily solvable. For suggestions on how to change your formulation, see Williams [1].

After you run the 'legacy' intlinprog algorithm once, you might want to change some options and rerun it. The changes you might want to see include:

  • Lower run time

  • Lower final objective function value (a better solution)

  • Smaller final gap

  • More or different feasible points

Here are general recommendations for option changes that are most likely to help the solution process. Try the suggestions in this order:

  1. For a faster and more accurate solution, increase the CutMaxIterations option from its default 10 to a higher number such as 25. This can speed up the solution, but can also slow it.

  2. For a faster and more accurate solution, change the CutGeneration option to 'intermediate' or 'advanced'. This can speed up the solution, but can use much more memory, and can slow the solution.

  3. For a faster and more accurate solution, change the IntegerPreprocess option to 'advanced'. This can have a large effect on the solution process, either beneficial or not.

  4. For a faster and more accurate solution, change the RootLPAlgorithm option to 'primal-simplex'. Usually this change is not beneficial, but occasionally it can be.

  5. To try to find more or better feasible points, increase the HeuristicsMaxNodes option from its default 50 to a higher number such as 100.

  6. To try to find more or better feasible points, change the Heuristics option to either 'intermediate' or 'advanced'.

  7. To try to find more or better feasible points, change the BranchRule option to 'strongpscost' or, if that choice fails to improve the solution, 'maxpscost'.

  8. For a faster solution, increase the ObjectiveImprovementThreshold option from its default of zero to a positive value such as 1e-4. However, this change can cause intlinprog to find fewer integer feasible points or a less accurate solution.

  9. To attempt to stop the solver more quickly, change the RelativeGapTolerance option to a higher value than the default 1e-4. Similarly, to attempt to obtain a more accurate answer, change the RelativeGapTolerance option to a lower value. These changes do not always improve results.

Some “Integer” Solutions Are Not Integers

Often, some supposedly integer-valued components of the solution x(intcon) are not precisely integers. intlinprog considers as integers all solution values within IntegerTolerance of an integer.

To round all supposed integers to be precisely integers, use the round function.

x(intcon) = round(x(intcon));

Caution

Rounding can cause solutions to become infeasible. Check feasibility after rounding:

max(A*x - b) % see if entries are not too positive, so have small infeasibility
max(abs(Aeq*x - beq)) % see if entries are near enough to zero
max(x - ub) % positive entries are violated bounds
max(lb - x) % positive entries are violated bounds

Large Components Not Integer Valued

intlinprog does not enforce that solution components be integer valued when their absolute values exceed 2.1e9. When your solution has such components, intlinprog warns you. If you receive this warning, check the solution to see whether supposedly integer-valued components of the solution are close to integers.

Large Coefficients Disallowed

intlinprog does not allow components of the problem, such as coefficients in f or ub, to exceed 1e20 in absolute value (1e25 for the "legacy" algorithm), or components of A or Aeq to exceed 1e15 in absolute value. If you try to run intlinprog with such a problem, intlinprog issues an error.

If you get this error, sometimes you can scale the problem to have smaller coefficients:

  • For coefficients in f that are too large, try multiplying f by a small positive scaling factor.

  • For constraint coefficients that are too large, try multiplying all bounds and constraint matrices by the same small positive scaling factor.

References

[1] Williams, H. Paul. Model Building in Mathematical Programming, 5th Edition. Wiley, 2013.