On the Generation of Cutting Planes which Maximize the Bound Improvement

We propose the bound-optimal cutting plane method. It is a new paradigm for cutting plane generation in Mixed Integer Programming allowing for the simultaneous generation of k cuts which, when added to the current Linear Programming relaxation, yield the largest bound improvement. By Linear Programming duality arguments and standard linearization techniques we show that, for a large family of cutting planes, the cut generating problem of our cutting plane method can be formulated as a Mixed 0-1 Integer Program. We highlight the potential of our technique by presenting computational experiments on the generation of bound-optimal stable set inequalities for the fractional clique number problem. We compare our method to the standard generation of maximally violated cuts, to the generation of cutting planes corresponding to a steepest-edge pivot in the dual, and to coordinated cutting plane generation. With respect to the standard algorithm, the bound-optimal cutting plane method allows for a substantial reduction in the number of cutting plane iterations and/or cuts needed to achieve either a given bound or an optimal solution. This reduction is still substantial also in comparison to the other techniques.