# MODELLI MATEMATICI PER L'OTTIMIZZAZIONE

Academic Year 2023/2024 - Teacher: Patrizia DANIELE

## Expected Learning Outcomes

The objectives of the course Mathematical Models for Optimization are as follows:

• to determine paths of minimum and maximum length starting from a root node;
• to apply the concepts of generalized derivatives to functions;
• to apply the Lagrange theory to constrained optimization problems;
• to formulate an equilibrium problem as an evolutionary variational inequality;
• to solve evolutionary variational inequalities.

Knowledge and understanding: the aim of the course is to be able in recognizing constrained optimization problems and in formulating real life problems in mathematical terms

Applying knowledge and understanding: students will be able to identify the functional characteristics of the data, to analyze various optimization situations, to propose optimal solutions to complex problems.

Making judgments: students will be able to analyze the data.

Communication skills: students will be able to communicate their experience and knowledge to other people.

Learning skills: students will have acquired the ability to learn, even autonomously, further knowledge on the problems related to applied mathematics.

## Course Structure

The course will be taught through lectures and exercises in the classroom and at the computer labs.

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.

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

Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco.

## Required Prerequisites

The basic concepts of Calculus I and II (differentiability, convexity of sets and functions, topology, ...), of Operations Research (concept of network and variational inequality) and of Optimization (minimum problems, sub-differentials, ...) are required

## Attendance of Lessons

Attendance is strongly recommended.

## Detailed Course Content

Graph theory (about 12 hours):

Graphs and digraphs: Definitions and preliminary notions, associated matrices. Kruskal's algorithm and its variant. Dijkstra's algorithm and its variant. Ford algorithm. Bellman-Kalaba’s algorithm. The traveling salesman problem.

Generalized derivatives (about 10 hours)

Directional derivatives, Gâteaux and Fréchet derivatives. Subdifferential.

Computational methods (about 8 hours)

The subgradient method. The discretization method.

Network models (about 17 hours)

Traffic networks. The Braess' paradox. Efficiency measure of a network. Supernetworks with three levells of decision-makers.

## Textbook Information

1. L. Daboni, P. Malesani, P. Manca, G. Ottaviani, F. Ricci, G. Sommi, “Ricerca Operativa”, Zanichelli, Bologna, 1975.
2. P. Daniele, “Dynamic Networks and Evolutionary Variational Inequalities", Edward Elgar Publishing, 2006.
3. J. Jahn, "Introduction to the Theory of Nonlinear Optimization", Springer, 1996.
4. Papers on STUDIUM  http://studium.unict.it

## Course Planning

SubjectsText References
1Paths of minimal and maximal length1
2Generalized derivatives and properties3
3Subdifferential of a function and properties3
4Traffic networks with additional constraints2
5 Latora-Marchiori and Nagurney-Qiang measures3

## Learning Assessment

### Learning Assessment Procedures

The final exam consists of an oral test during which the candidate demonstrates that he/she has assimilated the topics covered in the course.

Verification of learning can also be carried out electronically, should the conditions require it.

Final grades will be assigned taking into account the following criteria:

Rejected: Basic knowledges have not been acquired. The student is not able to solve simple exercises.

18-23: Basic knowledges have been acquired. The student solves simple exercises and has sufficient communications skills and making judgements.

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

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

### Examples of frequently asked questions and / or exercises

The algorithm for the shortest path in a graph.