A feasible rounding approach for granular optimization problems

We introduce a new technique to generate good feasible points of mixedinteger nonlinear optimization problems which are granular in the sense that a certain inner parallel set of their continuously relaxed feasible set is consistent. The latter inner parallel set was employed in O. Stein, Error bounds for mixed integer linear optimization problems, Mathematical Programming, Vol. 156 (2016), 101–123, as well as O. Stein, Error bounds for mixed integer nonlinear optimization problems, Optimization Letters, Vol. 10 (2016), 1153–1168, to bound the error occurring when an optimal point of the relaxed problem is rounded to the next integer point. On the contrary, in the present paper we show that efficiently solving certain purely continuous optimization problems over the inner parallel set and rounding their optimal points leads to feasible points of the original mixed-integer problem. For their objective function values we present computable a-priori and a-posteriori bounds on the deviation from the optimal value, as well as computable certificates for the granularity of a given problem. Computational examples for large scale knapsack problems and for several problems from the MIPLIB libraries illustrate that our method is able to outperform standard software. A post processing step to our approach, using integer line search, further improves the results.