Ateneo - UPO

Introduction to Nonlinear programming, Lagrange multipliers method. Vector spaces, bases, dimension. Eigenvalues and eigenvectors of square matrices and applications.

Notes by the teacher.Simon C.P -Blume L.E., Matematica 2 per l'Economia e le Scienze sociali, Università Bocconi Editore, 2002 (Cap. 13).

This course is worth 2 ECTS credits (CFU), corresponding to 15 hours of direct instruction. Overall, approximately 8 hours consist of delivered teaching (DE) and 7 hours of interactive teaching (DI), with flexible proportions within each lesson.
The course aims to help students acquire fundamental knowledge and practical skills related to basic concepts of non-linear programming and linear algebra, along with the ability to use these concepts in analyzing specific models.

First and second partial derivatives and their application to solving unconstrained optimization problems (Mathematical Methods 2);square matrices and their determinant, rank of a matrix, solving linear systems (Mathematical Methods 2).

Lectures aim to present the course topics (knowledge) and foster their critical discussion, adopting an integrated approach that combines direct instruction and interactive teaching. The course alternates theoretical sessions—preceded by motivating examples—with practical exercises designed to apply the introduced concepts. For each topic, exercises and questions are presented and discussed collectively in class to encourage everyone's participation and engagement.
Digital tools (the DIR platform, Wooclap) are used to support both classroom lectures and individual study.

Any useful information on the course and supplementary didactic resources can be found on the web page of the course at www.dir.uniupo.it.
Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/services- students-physical-or-learning-disabilities. Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The exam consists of a written test, comprising:
a question to assess the level of knowledge achieved regarding the concepts covered;an exercise to assess the knowledge and understanding of the course content, the ability to apply this knowledge, and the skill to express the results obtained using sufficiently rigorous technical language, regarding the solution of a non-linear programming problem;an exercise to assess the knowledge and understanding of the course content, the ability to apply this knowledge, and the skill to express the results obtained using sufficiently rigorous technical language, regarding the calculation of eigenvalues and eigenvectors of a square matrix.
An additional final exam session is scheduled at the end of lectures, conducted in the same manner as the official exam sessions.
Registration for the written exam via the student portal is mandatory: once the maximum number of exam attempts permitted by the Academic Regulations is reached, the system will not allow further registrations.

NONLINEAR PROGRAMMINGOptimization problems with equality constraints;substitution and level set methods, norm, distance, and inner product of vectors;Lagrange multiplier method;sufficient optimality conditions;applications: portfolio selection model, principal components of a random vector.
EIGENVALUES AND EIGENVECTORS OF A SQUARE MATRIXdefinition, existence, and computation of eigenvalues;characteristic polynomial, characteristic equation, algebraic multiplicity of an eigenvalue;eigenvalues of transpose and inverse, eigenvalues of triangular matrices;relationship between eigenvalues, trace, and determinant;computation of eigenvectors, geometric multiplicity of an eigenvalue;computation of eigenvalues and eigenvectors of symmetric matrices;similar matrices: definition, equality of eigenvalues;diagonalization of a square matrix.

KNOWLEDGE AND UNDERSTANDING
Sufficient/good knowledge of the main concepts of constrained optimization, eigenvalues, and eigenvectors.

APPLYING KNOWLEDGE AND UNDERSTANDING
Sufficient/good ability to solve simple equality-constrained optimization problems, and to calculate eigenvalues and eigenvectors in manually tractable cases.

MAKING JUDGEMENTS
Sufficient/good judgment in choosing and using appropriate computational tools to solve the aforementioned problems.

COMMUNICATION SKILLS
Sufficient/good ability to communicate clearly and rigorously, in written form, the logical-deductive steps followed in addressing the problems.

LEARNING SKILLS
Sufficient ability to independently use reference textbooks, lecture notes, and supporting digital resources to review and understand the solutions to standard problems and exercises, along with a basic ability to independently explore concepts not explicitly covered in class and apply them to relevant economic-financial models.