### Duration

30h Th, 20h Pr, 25h Proj.

### Number of credits

### Lecturer

### Language(s) of instruction

English language

### Organisation and examination

Teaching in the first semester, review in January

### Schedule

### Units courses prerequisite and corequisite

Prerequisite or corequisite units are presented within each program

### Learning unit contents

In a large number of engineering problems, many decisions can be undertaken leading to different solutions, some of them being more interesting than others. A way to decide on the best decision is to come up with a mathematical model in which all decisions are variables and the choice is made by considering a function of the values of all variables.

This formalism modeling many real-life problems is called mathematical programming. In a mathematical program, we define a set of decision variables, constraints linking the variables and defining what is a feasible solution and finally an objective function to optimize. Depending on the properties of all the considered functions, the obtained optimization problem can be more or less difficult to solve. In this course we consider three types of optimization problems: linear problems and their structure (duality), nonlinear problems that keep the nice structure (conic problems) and finally problems without any structure.

The following concepts are studied in the course:

- The revised Simplex Algorithm

- Duality for linear programming

- Post-optimal analysis and the Dual Simplex Algorithm

- Introduction to interior point methods

- Optimality conditions for nonlinear programs

- Conic programming and duality

- Numerical methods for nonlinear optimization

This course is given in English.

### Learning outcomes of the learning unit

At the end of the course, the student will be able to

- formulate a real problem in terms of a mathematical optimization model
- determine the complexity of an optimization problem and in particular whether it can be solved in polynomial time
- write the dual of a linear or a conic problem
- apply or implement the main optimization algorithms (simplex, dual simplex, interior-point methods, gradient descent, quasi-Newton)

This course contributes to the learning outcomes I.1, I.2, II.1, II.2, III.1, III.2, III.3, III.4, IV.1, IV.4, VI.1, VI.2, VI.3, VII.2, VII.3, VII.4, VII.5 of the MSc in data science and engineering.

This course contributes to the learning outcomes I.1, I.2, II.1, II.2, III.1, III.2, III.3, III.4, IV.1, VI.1, VI.2, VI.3, VII.2, VII.3, VII.4, VII.5 of the MSc in electrical engineering.

This course contributes to the learning outcomes I.1, I.2, II.1, II.2, III.1, III.2, III.3, III.4, IV.1, IV.3, VI.1, VI.2, VI.3, VII.2, VII.3, VII.4, VII.5 of the MSc in computer science and engineering.

This course contributes to the learning outcomes I.1, I.2, II.1, II.2, III.1, III.2, III.2, III.3, III.3, III.4, IV.1, VI.1, VI.2, VI.3, VII.2, VII.3, VII.4, VII.5 of the MSc in engineering physics.

### Prerequisite knowledge and skills

Basic course in linear algebra and calculus. Some basic knowledge of a programming language is also required.

### Planned learning activities and teaching methods

Traditional tutorials are organized for roughly 20 hours. A larger project consisting in modeling and solving a real-world problem using a linear programming package is also organized.

### Mode of delivery (face to face, distance learning, hybrid learning)

Face-to-face course

*Additional information:*

The lecture is given in face-to-face.

### Recommended or required readings

D. Bertsimas, J. Tsistsiklis. Introduction to linear optimization, Dynamic Ideas, 1997.

S. Boyd, L. Vandenberghe. Convex Optimization, 2004.

**Exam(s) in session**

Any session

- In-person

written exam ( open-ended questions )

**Written work / report**

*Additional information:*

The exam is written.

It is made of one theory question (a true/false with justification) and exercises similar to those solved in the tutorial sessions.

For the final grade, the exam mark counts for 2/3 and the project mark counts for 1/3. In case it is more advantageous for the student, the exam mark can count towards the entire grade. The project must be presented during the first session within the period provided for this purpose. There is no possibility to represent the project at another time.

### Work placement(s)

### Organisational remarks and main changes to the course

The course is taught in English.

All documents are avilable via ecampus.

### Contacts

The professor is Quentin Louveaux q.louveaux@uliege.be

The teaching assistants are Adrien Bolland adrien.bolland@uliege.be and Laurie Boveroux Laurie.Boveroux@uliege.be