General Prerequisites: Prelims Optimisation is recommended.
Course Overview: In many areas of practical importance linear optimisation problems occur with integrality constraints imposed on some of the variables. In optimal crew scheduling for example, a pilot cannot be fractionally assigned to two different flights at the same time. Likewise, in combinatorial optimisation an element of a given set either belongs to a chosen subset or it does not. Integer programming is the mathematical theory of such problems and of algorithms for their solution. The aim of this course is to provide an introduction to some of the general ideas on which attacks to integer programming problems are based: generating bounds through relaxations by problems that are easier to solve, and branch-and-bound.
Lecturer(s):
Prof. Raphael Hauser
Learning Outcomes: Students will understand some of the theoretical underpinnings that render certain classes of integer programming problems tractable ("easy'' to solve), and they will learn how to solve them algorithmically. Furthermore, they will understand some general mechanisms by which intractable problems can be broken down into tractable subproblems, and how these mechanisms are used to design good heuristics for solving the intractable problems. Understanding these general principles will render the students able to guide the modelling phase of a real-world problem towards a mathematical formulation that has a reasonable chance of being solved in practice.
Course Synopsis: Course outline. What is integer programming (IP)? Some classical examples.
Further examples, hard and easy problems.
Alternative formulations of IPs, linear programming (LP) and the simplex method.
LP duality, sensitivity analysis.
Optimality conditions for IP, relaxation and duality.
Total unimodularity, network flow problems.
Optimal trees, submodularity, matroids and the greedy algorithm.
Augmenting paths and bipartite matching.
The assignment problem.
Dynamic programming.
Integer knapsack problems.
Branch-and-bound.
More on branch-and-bound.
Lagrangian relaxation and the symmetric travelling salesman problem.
Solving the Lagrangian dual.
Branch-and-cut.