Improving the LP bound of a MILP by branching concurrently

In this paper the branching trees for attacking MILP are reviewed. Under certain circumstances branches can be done concurrently. This is fully investigated with the result that there are restrictions for certain dual values and reduced costs. As a sideeffect of this study a new class of cuts for MILP is found, which are defined by those values. 1 Motivation of the following thoughts Nowadays the technique for doing MILP (Mixed Integer Linear Programming) is based on the branch and bound method. This method uses the best solution of the linear inequality system with objective function (= LP-instance) by leaving out the integer conditions from the mixed integer linear inequality system with objective function (= MILP-instance). Then this method searches for an 0-1 (or integer) variable xn, which has a non-integer value qn. The next step is to create two new LP-instances by adding first xn = 0 (or xn ≤ [qn]) and secondly xn = 1 (xn ≥ [qn + 1]). By continuing this process a binary tree of problems is created. Now take two different nodes in this tree, so you look at two different LPinstances. With both problems it is possible that some xn is still not integer. We’ll create a branch on that variable for both problems. It can happen at a big and sparse MILP-instance, that the same similar branching will lead to exactly the same calculations at the new LP-instances. From a numerical point of view this is unsatisfactory. New ideas have been developed here which use some kind of independence of branching. These will help to prevent such double calculations. One further aim of these new techniques is a better measurement and control of what happens at a branching. A practical and short-term outcome should be better limits for