# 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.

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