Course Overview: In many practical problems that can be approached via linear optimisation problems, some or all of the variables are constrained to take binary or integer values. For example, in optimal crew scheduling 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.

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 enable the students 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: Week 1:

Classical examples of Integer Programming problems (IP), modelling and basic terminology.

Linear programming I: the simplex method.

Week 2:

Linear programming II: Duality Theory.

Total unimodularity I: Ideal formulations of IPs and totally unimodular matrices.

Week 3:

Total Unimodularity II: Exact theoretical characterisation, practical sufficient criteria, bipartite matching, the shortest path problem.

Submodularity I: Submodular functions and submodular optimisation problems.

Week 4:

Submodularity II: Submodular rank functions, matroids, the greedy algorithm and the maximum weight independent set problem.

Branch-and-Bound I: LP based branch-and-bound for general integer programming problems.

Week 5:

Branch-and-bound II: general B&B, pre-processing, warm starting of LPs, dual simplex method.

Dantzig-Wolfe decomposition, delayed column generation.

Week 6:

Branch-and-Price, application to the cutting stock problem.

Preprocessing of LPs and IPs, generating valid cuts, cutting plane algorithm.

Week 7:

Chvatal cuts, Gomoroy cuts, branch-and-cut algorithm.

The Generalised Assignment Problem.

Week 8:

Lagrangian relaxation and Lagrangian duality.

The subgradient algorithm.