Continuous optimization is more difficult than discrete optimization

Integer optimization

Allowable inequalities can be added to the model statically or dynamically as needed. In the case of dynamic procedures cuts specifically used to separate fractional values ​​from variables that are subject to the integer condition in a B&B process. These include cutting planes (engl .: cutting plane methods) and Branch & Cut methods (B&C), which can now be found in some commercial software packages.

Section plane methods were introduced by Gomory in 1958 and are based on the concept of the one introduced above cuts. The B&C The algorithm combines the basics of the O&M method and the cutting plane method and proceeds in a similar way to the O&M method, but with each step further cuts added to successively exclude fractional values ​​of the variables. Here, too, it is a question of a tree search procedure, but in addition to or instead of branching to a certain variable, another cut is added. B&C works in two different ways. The simple procedure will only be cuts added in terms of admissible inequalities; this may turn out to be very many, but not necessarily very effective cuts added. In the second method, only inequalities (cuts) are added which represent a permissible inequality with regard to a local node (because branches have already taken place), but not with regard to the whole problem; In doing so, particular care is taken to add a cut that is actually violated by the assumption of fractional values; adding it causes fractional values ​​to be excluded. Be meaningful cuts, which are no longer effective, are also removed again dynamically. In addition to the cuts, which can be constructed by an experienced modeler, commercial MILP software now offers the possibility of checking whether a model has been subjected to certain standard cuts, e.g. B. Gomory cuts, can be improved.