## MA306-6-AU-CO:Combinatorial Optimisation

The details
2024/25
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Current
Thursday 03 October 2024
Friday 13 December 2024
15
18 March 2024

Requisites for this module
MA114 and MA205
(none)
(none)
(none)

(none)

(none)

## Module description

Combinatorial optimization is one of most active areas of discrete mathematics and deals with optimization in domains where (some or all of) the variables are integral. The main motivation is that a plethora of real-life problems can be modelled as abstract combinatorial optimization problems. The module focuses on problems that can be formulated as integer or mixed integer linear programs and on the algorithms one can apply to solve such programmes, as well as on the underlying theory.

## Module aims

The aims of this module are:

• To introduce classic discrete optimization problems, like scheduling, the traveling salesman problem, and set covering, through the lens of integer linear programming.

• To present polyhedral theory and equip the students with the tools to understand the underlying connections between the mathematical theory and the algorithms used to solve integer linear programs.

• To provide students with an understanding of the main algorithmic techniques for solving (mixed) integer linear programs, as these are often the building blocks of the state-of-the-art approaches used today to deal with massive instances.

## Module learning outcomes

By the end of the module, students will be expected to:

1. formulate a wide range of combinatorial optimization problems as integer programs;

2. describe feasible sets as polyhedra using facets, extreme points and extreme rays;

3. identify and generate valid inequalities for feasible sets;

4. use linear programming relaxation and duality to generate bounds for integer programs' objective values;

5. solve integer programs with cutting-plane algorithms;

6. solve integer and mixed integer program with Branch-and-Bound;

7. apply Benders' decomposition algorithm to mixed integer programs.

## Module information

Indicative syllabus

• Scope of integer and combinatorial programming: Modelling with integer variables.

• Pre-processing: Balas’s constraint disaggregation procedure.

• Polyhedral theory: Valid inequalities; Facet constraints; Convex hull of integer solutions.

• LP relaxation of integer programming problems.

• General integer programming algorithms: Cutting plane algorithms, Branch-and-Bound.

• Special purpose algorithms for Mixed Integer Programming: Benders decomposition.

## Learning and teaching methods

Teaching in the department will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography

The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.

## Assessment items, weightings and deadlines

Coursework / exam Description Deadline Coursework weighting
Coursework   Assignment 1
Coursework   Assignment 2
Exam  Main exam: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period)
Exam  Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment Period)

### Exam format definitions

• Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
• In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
• In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
• In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Coursework Exam
20% 80%

### Reassessment

Coursework Exam
20% 80%
Module supervisor and teaching staff
Dr Georgios Amanatidis, email: georgios.amanatidis@essex.ac.uk.
Dr Georgios Amanatidis
georgios.amanatidis@essex.ac.uk

Availability
Yes
Yes
No

## External examiner

Dr Yinghui Wei
University of Plymouth
Dr Murray Pollock
Newcastle University
Director of Statistics / Senior Lecturer
Resources
Available via Moodle
Of 30 hours, 28 (93.3%) hours available to students:
2 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.

Further information

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