# 69441 - OPTIMIZATION MODELS AND ALGORITHMS M

## Learning outcomes

The goal of the course is to deal with Integer Programming that is a very powerful tool for modeling combinatorial optimization problems arising in many branches of engineering, industry and resource allocation. The first part of the course covers the modeling aspects of the field, providing the tools for constructing effective mathematical models, i.e., models that can be solved in practice. The second part is devoted to the algorithmic aspects: basic algorithms are reviewed and more sophisticated ones, useful for those models characterized by a large number of variables and/or constraints, are presented in detail. Finally, the third part of the course discusses real-world applications. At the end of the course students are able to formalize a combinatorial problem taken for the real life and run specific tools and algorithms for solving it in practice.

## Course contents

Requirements: a good knowledge of linear algebra and basic mathematics is required.

The course is given in English: slides and exercises are in English. The exam must be taken in English.

The course focuses on optimization problems arising in decision making and the definition of mathematical models and exact algorithms to solve them. The mathematical theory on which these methods are based is elaborated through the study of theorems.

Contents

• Introduction to optimization problems arising in decision making and to Mathematical Programming
• Convex programming, Linear Programming, simplex algorithm, duality theory
• Integer Linear Programming, branch-and-bound algorithm, classical combinatorial optimization problems, relaxations, computational complexity
• Exponential-size models, column generation, constraint separation
• Graph problems, examples of real-life applications, optimization software tools

Slides on virtuale.unibo.it (at the web page of the course)

Fischetti M. Introduction to Mathematical Optimization. Kindle Direct Publishing, 2019.

Christos H. Papadimitriou and Ken Steiglitz, Combinatorial optimization: algorithms and complexity. Dover, 1998.

D. Bertsimas and J. Tsitsiklis, Introduction to linear programming. Dynamic Ideas and Athena Scientific, Belmont, Massachusetts, 2008.

D. Bertsimas, D. and R. Weismantel, Optimization over integers. Dynamic Ideas, Belmont, Massachusetts, 2005.

## Teaching methods

The course consists of lectures and class exercises.

## Assessment methods

The exam consists of a written exam in which some exercises on topics of the course have to be solved (about 60 minutes). On the day after the written exam, an oral exam covering all topics of the course (including theorems and proofs) completes the exam.

Written and oral exams must be done in the same exam round. After the written exam, the list of students that are admitted to the oral exam is provided. There is a unique mark after the oral exam.

## Teaching tools

Slides on virtuale.unibo.it (at the web page of the course) and optimization software tools

## Office hours

See the website of Valentina Cacchiani