Mathematical optimizacion

The main objective of this subject is to introduce students to the basic concepts and techniques of mathematical optimization.
As more specific objectives, students must:

- Know how to identify and model mathematical optimization problems.
- Know how to solve mathematical optimization problems using the appropriate techniques and algorithms.
- Know and identify the structure and properties of mathematical optimization problems.
- Be familiar with the interrelations between mathematical optimization and machine learning.

TOPIC 1. INTRODUCTION TO MATHEMATICAL OPTIMIZATION
TOPIC 2. MODELLING AND PRACTICAL RESOLUTION OF OPTIMIZATION PROBLEMS.
TOPIC 3. LINEAR PROGRAMMING
TOPIC 4. INTEGER PROGRAMMING
TOPIC 5. NETWORK OPTIMIZATION PROBLEMS
TOPIC 6. GENERAL FOUNDATIONS OF CONSTRAINED NONLINEAR OPTIMIZATION
TOPIC 7. OPTIMIZATION AND MACHINE LEARNING

Basic bibliography
- Bazaraa, M., Jarvis, J., Sherali, H. (2010). Linear Programming and Network Flows. Wiley and Sons.
- Hillier, F., Lieberman, G. (2002). Investigación de operaciones. McGraw-Hill.

Complementary bibliography
- Ahuja, R. K., Magnanti, T. L., Orlin, J. B. (1993). Network Flows. Theory, Algorithms and Applications. Prentice-Hall.
- Bazaraa, M., Jarvis, J., Sherali, H. (2005). Programación lineal y flujo en redes. Limusa.
- Bazaraa, M. S., Sherali, H. D., Shetty, C. M. (2013). Nonlinear programming: theory and algorithms. John Wiley & Sons.
- Boyd, S., Vandenberghe, L. (2004). Convex optimization. Cambridge university press.
- Hillier, F.S., Lieberman, G. J. (2006). Introducción a la Investigación de Operaciones. McGraw-Hill.
- Papadimitriou, C. H., Steiglitz, K. (1998). Combinatorial optimization: algorithms and complexity. Courier Corporation.
- Winston, W. (2004). Investigación de Operaciones. Paraninfo.
- Salazar González, J.J. (2001). Programación Matemática. Díaz de Santos.
- Taha, H. A. (2004). Investigación de Operaciones. Pearson, Prentice Hall.

This subject will work on the basic, general and transversal competences contained in the memory of the title of bachelor’s degree in Artificial Intelligence. The competences that will be specifically enhanced in this subject are listed below:

CE1 - Ability to use mathematical and statistical concepts and methods to model and solve artificial intelligence problems.
CE5 – Ability to understand and apply the basic principles and techniques of parallel and distributed programming for the development and efficient execution of artificial intelligence techniques.
CE15 – Knowledge, correct application, and explanation of the validation techniques of artificial intelligence solutions.
CG2 - Ability to solve problems with initiative, decision-making, autonomy, and creativity.
CG4 - Ability to select and justify the appropriate methods and techniques to solve a specific problem, or to develop and propose new methods based on artificial intelligence.
CB2 - Ability to apply their knowledge to their work or vocation in a professional manner and possess the competences that are usually demonstrated through the elaboration and defence of arguments and the resolution of problems within their area of study.
CB5 – Ability to manage those learning skills necessary to undertake further studies with a high degree of autonomy.
TR3 - Ability to create new models and solutions autonomously and creatively, adapting to new situations. Initiative and entrepreneurial spirit.

Lectures (30 hours). For the transfer of knowledge, slides and blackboard will be used and standard problems will be solved, so that students can work on the exercise sheets provided. As for the material to follow the subject, in addition to the recommended bibliography, students will have the help of material on the Virtual Campus of the USC. In the expository teaching sessions, the following skills will be worked on basic skills (CB2 and CB5), general skills (CG2 and CG4), transversal skills (TR3) and subject-specific skills (CE1, CE5, and CE15).

Lab sessions (20 hours). In this type of teaching, practical exercises will be solved, emphasising the practical application of theoretical concepts, and favouring the use of specific software for solving optimisation problems. Objectives developed: basic skills (CB2 and CB5), general skills (CG2 and CG4), transversal skills (TR3) and subject-specific skills (CE1, CE5, and CE15).

Tutorials (2 hours). The tutorials will be aimed at monitoring student learning. Activities will be carried out to achieve an overview of the subject as a whole and to identify the aspects that need to be improved. Objectives developed: basic skills (CB2 and CB5), general (CG4), transversal (TR3) and subject-specific skills (CE1).

The distribution of the hours of lectures (30 hours) and labs (20 hours), by topic, is as follows:

Topic 1. Introduction to Mathematical Optimization (2 lectures and 2 labs).
Topic 2. Modelling and practical resolution of optimization problems (2 lectures and 2 labs).
Topic 3. Linear programming (8 lectures and 4 labs).
Topic 4. Integer programming (3 lectures and 2 labs).
Topic 5. Network optimization problems (5 lectures and 4 labs).
Topic 6. General foundations of constrained nonlinear optimization (5 lectures and 2 labs).
Topic 7. Optimization and machine learning (5 lectures and 4 labs).

The course material will be made available to students through the USC Virtual Campus.

During the course, the degree to which students have achieved the objectives proposed for this subject will be continuously assessed, ending with a theoretical-practical exam. The weighting of each part of the assessment is detailed below.

Continuous evaluation (30%): continuous evaluation will be based on participation in different types of tasks. Continuous evaluation activities will include the resolution of practical cases (individually or in groups), which may include the use of specific software. Exercises for individual resolution will also be proposed, which will be carried out in a face-to-face and/or non-face-to-face manner. The mark obtained will be maintained between opportunities in the same academic year (ordinary and extraordinary). The participation and involvement of students in the classroom will also be assessed during this part of the course. To be eligible for continuous evaluation, whenever possible, it is necessary to attend at least 75% of the practical sessions. Competences assessed: CE1, CE5, CE15, CG2, CG4, CB2, CB5 and TR3.

Final exam (70%): the final exam will consist of several questions and theoretical-practical exercises that will deal with the contents of the subject, which may include the interpretation of the results obtained with the software used in interactive teaching. Competences assessed: CE1, CE5, CE15, CG2, CG4, CB2, CB5 and TR3.

The weight of the continuous evaluation in the second opportunity will be the same as in the ordinary call for the semester. For repeating students, the evaluation will be carried out in the same manner, and no grades obtained in the previous course will be retained (including the continuous assessment grade).

The student will have the condition of "no-show" when he/she does not attend the theoretical exam and has not attended to any of the continuous evaluation activities.

In cases of fraudulent performance of exercises or tests, the provisions of the “Regulations for the evaluation of student academic performance and the review of grades” will apply.

In this subject, students have the following teaching provided by the teaching staff: 30 hours of lectures and 20 hours of laboratories. It is recommended that two hours of additional work be devoted to each hour of lecture and laboratory classes, in addition to the hours of tutorials. During these hours, the knowledge acquired should be deepened through the revision of concepts, the practice of problem solving and the consultation of the recommended bibliography. Thus, the approximate autonomous work of the student is 99 hours and hour of tutorial.

Following the lectures and interactive sessions is essential in order to pass the subject. Students must complete all the activities recommended by the teaching staff (problem solving, literature review and practical exercises) in order to successfully pass the subject. In addition, it is recommended to make use of the tutorial hours to resolve any doubts that may arise. It is also recommended to have taken the courses "Algebra", "Calculus and numerical analysis" and "Discrete Mathematics".

This guide and the criteria and methodologies described in it are subject to modifications derived from USC regulations and guidelines.