METODI MATEMATICI PER L'OTTIMIZZAZIONE

The course aims to present well-known optimization methods. The course provides students with the analytic tools to model and solve numerical situations in which a single decision-maker has to find the best choice. The attention focuses on applications in economics, engineering, and computer science.

Learning outcomes:

Students should be able to classify optimization problems according to their mathematical properties.

Students should be able to perform a theoretical investigation of a given optimization problem in order to assess its complexity.

Students should be able to write down first and second-order optimality conditions.

Students should be able to solve simple optimization problems without a computer.

Students should be able to solve optimization problems in a computer environment.

Students should be able to analyze the obtained solutions.

The goals of the course are:

To acquire base knowledge that allows students to study optimization problems and apply opportune techniques to solve decision-making problems. The students will be able to use algorithms for nonlinear programming problems.

To identify and model real-life decision-making problems. In addition, through real examples, the student will be able to find correct solutions for complex problems.

To choose and solve autonomously complex decision-making problems and to interpret the solutions.

To acquire basic communication and reading skills using technical language.

To provide students with theoretical and practical methodologies and skills to deal with optimization problems, ranging from computer science to engineering; to acquire further knowledge on the problems related to applied mathematics.

Teaching Organization

credit value 6 - 47 hours

total study 150 hours

103 hours of individual study
35 hours of frontal lecture

12 hours of exercises

For this course, there will be 2 hours of teaching per lecture twice a week. The course includes classroom lessons and exercises. Teaching material will be available on Studium and Teams platforms. For each topic, exercises will be solved by the teacher or proposed to students.
 
Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

No requirements.

Course attendance is strongly recommended.

The course will cover the following topics:

Fundamentals concepts (6 hours). One-dimensional optimization: convex and quasiconvex functions, first-order methods, local and global minima. Existence of solutions: continuous and lower semicontinuous functions, coercive functions, Weierstrass theorem, unique and nonunique solutions.

Optimality conditions (15 hours). Theory of optimality conditions: Fermat principle, the Hessian matrix, positive and negative semidefinite matrices, the Lagrange function and Lagrange multipliers, tangent cones, the Karush-Kuhn-Tucker conditions, regularity, complementarity constraints, stationary points.

Methods (6 hours). First-order optimization methods: the steepest descent method, conjugate directions, and gradient-based methods. Second order optimization methods: Newton's method and modifications.

Nondifferentiable optimization (3 hours). Subgradients, subdifferential and methods.

Multiobjective optimization (5 hours). Pareto frontiers and solution methods.

Exercises (12 hours)

Analytical solutions. Use of software (GeoGebra, Matlab, Mathematica).

Goals of U. N. Agenda for Sustainable Development

This course contributes to the achievement of the following goals of U. N. Agenda for Sustainable Development

Goal N. 4 Quality Education

      Target 4.3

     Target 4.7

Goal N. 13  Climate Action

                       Target  13.3

Method: classroom lesson

[P] Patriksson et al., An Introduction to Optimization: Foundations and Fundamentals Algorithms, Dover Publications Inc., 2019
[1] R. T. Rockafellar, R. J-B Wets, Variational Analysis
[2] S. Boyd, L. Vandenberghe, Convex optimization
[3] J. Jahn, Introduction to the Theory of Nonlinear Optimization - Springer- Verlag, Berlin (1996)
[4] F.S. Hillier, G.J. Lieberman, Introduction to Operations Research, Mc Graw Hill, 2020
Teaching material will be given during the course.
 

The final exam consists of an oral test during which the candidate may be asked to solve a numerical exercise. The final grade is established on the basis of the answers given by the candidate and the solving of the numerical example.

Rejected: Basic knowledge has not been acquired. The student is not able to solve simple exercises.

18-23: Basic knowledge has been acquired. The student solves simple exercises, has sufficient communication skills, and makes judgements.

24-27: All the knowledge has been acquired. The student solves all the proposed exercises making few errors and has good communication skills and making judgements.

28-30 cum laude: All the knowledge has been completely acquired. The student applies knowledge and has excellent communication skills, learning skills and making judgements.

Learning assessment may also be carried out on-line, should the conditions require it.

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipata — Servizi per le Disabilità e/o i DSA) referring teacher within DMI (https://web.dmi.unict.it/it/corsi/lm-18/referente-cinap).

To take part in the final exam, students must register through the SmartEdu portal. For any technical issues related to the registration, please contact the Student Office.