# Research

Dr. Richard’s research focuses on the design and application of methods for the global solution of linear and nonlinear mixed-integer optimization problems. Integer Programming is a branch of optimization that considers problems in which some of the variables are required to be integer-valued. These problems are typically very hard to solve to provable optimality. However, they can be used to model various systems with the view of making them more efficient. As a result, they are omni-present in Engineering and Management. In particular, they have found successful applications in field as diverse as forestry, mining and finance. From a practical point of view, theoretical advances in this field are significant as they could raise significantly the current state of the art of optimization solvers. In turn such advances would help many O.R. practitioners across all fields to solve, in a reasonable amount of time, many practical problems that are currently intractable and therefore improve the efficiency of many systems. Dr. Richard is interested in both the theoretical and practical sides of Mixed Integer Programming.

**Cutting Planes Algorithms for Linear Mixed Integer Programs:**

Although Linear Mixed Integer Programs form an apparently simple extension of linear programming, they are typically very hard to solve. Dr. Richard’s main research interests are in the development of new methods for solving these problems with a particular focus on methods that do not require the knowledge of the particular structure of the problem. Such a research angle allows the derivation of general tools that can be easily applied in commercial solvers and therefore can be easily included in the toolbox of OR practitioners. In particular, Dr. Richard is interested in the design of new lifting techniques for mixed integer programming that are more encompassing than traditional methods because they consider continuous variables and/or multiple problem constraints. Dr. Richard is also interested in uncovering relations that exist between lifting and existing approaches to integer programming, with the goal of generating a new generation of improved techniques. Finally, Dr. Richard is interested in how group theoretic approaches can be applied to generate new families of algorithms for Mixed Integer Programs.

**Methods for Nonlinear Mixed Integer Programs:**

Nonlinear Mixed Integer Programs find applications in virtually all fields of engineering as discrete variables can be used to represent design decisions about a system and nonlinearities naturally arise from the physical nature of the underlying system. The solution of these problems is typically very difficult. Dr. Richard is interested in determining how convex relaxations of nonlinear mixed integer programs can be built that are stronger than those currently available. Such convex relaxations can be used to obtain approximate solutions in many cases but, more importantly, can also be used to obtain global solutions provided that they are incorporated in a branch-and-bound framework. Dr. Richard is particularly interested in how lifting and disjunctive programming techniques can be used to generate these convex relaxations in the space of original variables.

**Practical Applications of Mixed Integer Programs:**

Dr. Richard has developed models for various practical applications. He worked on the use of optimization techniques for railcars assignment problems, water infrastructure design against intentional attacks and homeland security risk reduction problems. He also participated in the development of new MIP techniques for fluence map optimization problems arising in cancer radiation therapy. One of his main achievement include the development of a model for the assignment of empty freight car at a large US Railroad that was implemented by the company and has provided significant costs savings.

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## Problem Instances

“Lifted Tableaux Inequalities for 0-1 Mixed Integer Programs: A Computational Study” by A. K. Narisetty, J.-P. P. Richard and G. L. Nemhauser

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