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), presente in Biblioteca

Knowledge and understanding of some basic notions of nonlinear programming and linear algebra and ability to use these notions in the analysis of some models.

It is recommended to take the exam of Mathematical Methods 2.

Lectures, tutorials.

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.

A compulsory written exam consisting in: - a question designed to test the level of knowledge achieved of the concepts considered; - two exercises aimed to test the knowledge and understanding of the course contents, the ability to apply this knowledge and the ability to express the results obtained with a sufficiently rigorous technical language, with regard to the solution of a nonlinear programming problem and the calculation of eigenvalues and eigenvectors of a square matrix. There are 7 exam sessions in each solar year but students can enroll no more than three times, at their own choice. It is mandatory to register for the exam through the students portal: once the three exam participations have been reached, including withdrawals and refusals, the system will not allow further registrations.

Nonlinear programming: optimization problems with equality constraints. Substitution and level sets methods, Lagrange multipliers method. Optimal sufficient conditions. Applications: portfolio selection model, Principal Components of a random vector. Eigenvalues and eigenvectors of a real square matrix. Properties of eigenvalues, repeated eigenvalues and diagonalization of a square matrix. Eigenvalues and eigenvectors of symmetric, positive matrices.

Achieving a good knowledge and understanding of the notions of constrained optimum, eigenvalue and eigenvector. Development of a good ability to solve simple constrained optimization problems, to calculate eigenvalues and eigenvectors in cases that can be manually treated. Ability to apply the acquired knowledge to the study of some applied models.