# METODI MATEMATICI PER L'OTTIMIZZAZIONE

**Academic Year 2023/2024**- Teacher:

**Laura Rosa Maria SCRIMALI**

## Expected Learning Outcomes

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:

**Knowledge and
understanding**: 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.

**Applying knowledge and understanding**: 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.

**Making judgments**: to choose and solve autonomously
complex decision-making problems and to interpret the solutions.

**Communication skills**: to acquire base
communication and reading skills using technical language.

**Learning skills**: 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.

## Course Structure

**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. During the classroom lessons a graphics tablet will be used. The hand-written slides will be available. 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.

**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.

## Required Prerequisites

## Attendance of Lessons

#### Course attendance is strongly recommended.

## Detailed Course Content

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)

## Textbook Information

#### [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.

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Decision models | [4], teaching material |

2 | Convex sets, convex functions | [P], [1], [2], [3] teaching material |

3 | Cone, tangent cone, normal cone | [P], [1], [3], teaching material |

4 | Optimality conditions for unconstrained optimization | [P], [1], [2], teaching material |

5 | Optimality conditions for constrained optimization | [P], [1], [2], [3], teaching material |

6 | Duality | [P], [1], [2], [3] teaching material |

7 | Solution methods | [P], [4], teaching material |

## Learning Assessment

### Learning Assessment Procedures

The final exam consists of an oral test during which the candidate is also requested 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.

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

**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 online if the conditions should require it.

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