# Products of Variables in Mixed Integer Programming

## Products of Variables in Mixed Integer Programming

Products of problem variables appear naturally in quadratic programs. Special preprocessing, linearization and cutting plane techniques are available to deal with such products. If at least one of the two variables in a product is binary, then the product can be modeled using a set of linear constraints. As a consequence, there are many mixed integer linear programs (MILPs) that actually contain products of variables hidden in their constraint structure. Rediscovering these product relationships between the variables enables us to exploit the solving techniques for product terms.

## Webinar Summary

Products of problem variables appear naturally in quadratic programs. Special preprocessing, linearization and cutting plane techniques are available to deal with such
products.

If at least one of the two variables in a product is binary, then the product can be modeled using a set of linear constraints. As a consequence, there are many mixed integer linear programs (MILPs) that actually contain products of variables hidden in their constraint structure. Rediscovering these product relationships between the variables enables us to exploit the solving techniques for product terms.

In this webinar, we will:

• Explain how such product relationships can be detected in a given mixed integer linear program
• Demonstrate ideas on how they can be exploited to improve the performance of an MILP solver
• Describe cuts from the Reformulation Linearization Technique (RLT) and cuts for the Boolean Quadric Polytope (BQP)
• Present preliminary computational results from these techniques in Gurobi version 9.0

## Presenter

This webinar is presented by, Tobias Achterberg, Director of Development. Dr. Achterberg studied mathematics and computer science at the Technical University of Berlin and the Zuse Institute Berlin. He finished his PhD in mathematics under supervision of Prof. Martin Grötschel in 2007. Dr. Achterberg is the author of SCIP, currently the best academic MIP solver. In addition to numerous publications in scientific journals, he has also received several awards for his dissertation and for SCIP, such as the Beale-Orchard-Hays Prize.

## Presented Materials

You can download the materials associated with this webinar here.