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solver:implementing_new_branching_rules

# Branching on sums of binary variables

Q: I have implemented a MIP-model in GAMS, which I would like to solve to proven optimality. The problem is degenrate and has a lot optimal solutions, and many solutions very close to the optimum. It is not terminating with the optimal solution, but continuing to examine nodes with a very small gap > 0.00 %. I have a good idea for a branching rule, which i believe can work around the degeneracy problems. Instead of doing binary branching on fractional binary variables, I intend to branch on the sum of some variables, i.e:  sum(s, delta(s)) =l= k; or \sum(s, delta(s)) =g= k+1;

My question is then, how can I introduce such a branching rule in GAMS, instead of just using binary branching ??

You could introduce another integer variable sum_delta =e= sum(s, delta(s)); Now you can use branching priorities to instruct the MIP solver to first branch on the sum_delta variable and then on the delta(s):

delta.prior(s) = 100;
sum_delta.prior = 1;
mymodel.prioropt=1;
solve mymodel min obj using mip;

If the multiple close to optimal solutions come from symmetry, you might also want to try a more aggressive level for symmetry braking cuts (see e.g. cplex option symmetry). Preprocessing at the node or aggressive probing might also help (see e.g. Cplex options preslvnd and probe).