OPTIMIZATION

This compulsory course of 7 credit aims at introducing and developing many of the analytical tools which are generally used in theoretical and applied Economics.
In fact, the course aims to provide students with the basic principles of Mathematical Analysis which bear special relevance in the study of Mathematical Economics.

The instructional goal of the course is to provide students with a general understanding of the mathematical objects used in Economics, in order
- to develop knowledge about their properties and the relationships between them;
- to find efficient methods to solve problems, both abstract and concrete, in Unconstrained and Constrained Optimization;
- to find efficient methods to solve problems, both abstract and concrete, in Static and Dynamic framework.

Particular attention will be posed on the development of the analytical tools necessary for understanding and proving some of the basic results in Mathematical Economics that play a central role in Unconstrained and Constrained Optimization Theory, and in Static and Dynamic framework.
The course allows students to acquire knowledge for approaching the study of main economic problems where these tools are employed, both in static and in dynamic setting.

In particular, at the end of the course, the student is expected to be able of:
- knowing the theoretical results and the mathematical tools commonly used in the solution of Static or Dynamic Optimization problems;
- applying their knowledge and understanding, and problem solving abilities to formalize, discuss and solve Optimization problems;
- communicating the obtained results using a clear and appropriate mathematical notation and language, to specialist and nonspecialist audiences.

Students are expected to be familiar with standard material in Mathematics (at the level of courses taught to economics undergraduates). More precisely, the following topics are assumed to be very well-known: calculus in one variable; basic notions in calculus of several variables; basic notions in integral calculus; Linear Algebra: matrix and vector algebra, determinants, linear independence, basics of vector spaces, systems of linear equations, vector spaces and subspaces, subspaces attached to a matrix.
Reference: Chapters 2-5, Appendix A4, Chapters 7--14, 23, 26--28 of
Simon C.P., Blume L.E. (1994): Mathematics for Economists, W.W.Norton & Company Press, Cambridge, (1994).

The course program includes presentation of definitions and theoretical results, discussion and solution of problems on the following topics:
- Basic concepts of Topology of R^n
- Unconstrained Optimization.
- Constrained Optimization (equalities, inequalities).
- Concavity/convexity in Optimization.
- Metric, normed spaces. Contractions. Fixed points. Banach's Theorem.
- Parametric continuity: Maximum Theorems.
- Discrete time optimization: FHDP problems. Bellman Principle.
- Discrete time optimization: SDP problems. Bellman Principle. Existence result.

There is no single textbook for the class.
Before each meeting, in the moodle space detailed information on content and reading material (page numbers, selected exercises) will be made available to participants with reference to main textbooks.
Lecture notes, slides and exercises will be made available in the Moodle space of the course.

In alphabetical order:
Carter, M. (2001): Foundations of mathematical economics. MIT Press.
Iacopini M. (2016): Basics of Optimization Theory with Applications in MATLAB and R, Quaderni di didattica - Dip. di Economia, Ca' Foscari University of Venice
http://www.unive.it/pag/fileadmin/user_upload/dipartimenti/economia/doc/Pubblicazioni_scientifiche/quaderni_didattica/Quaderno_di_DIdattica_1_2016.pdf
Sundaram R.K. (1999): A First Course in Optimization Theory, Cambridge University Press, Cambridge.
Sydsaeter K., Hammmond P., Seierstadt A., Strom A. (2005): Further Mathematics for Economic Analysis, Prentice Hall.

In each academic year there are 4 exam sessions.
The first session takes place during the first semester (while the course is being taught), and it is articulated in a midterm partial written examination administered at the end of the first teaching period and a final partial written examination administered at the end of the second teaching period.
The final written evaluation is obtained as a weighted average of the two partial examinations and of attendance. The remaining sessions (from January until September) present a single comprehensive examination on the full contents of the course.

The objective of the examination is to test the student's ability to understand the problem set, to choose the most appropriate tools for solving the proposed problems, to rigorously apply the selected methods and to discuss the proposed solutions.
Written exams typically consist of exercises similar to those solved during the course, each exercise with its own score. In answering the exam no help can be used (books, notes, computers, calculators, web,…).

There will be thirty lectures in 10 weeks of teaching activities, 4 meetings each lasting 2 academic hours per week: three meetings with the main lecturer (i.e. 60 academic hours) and one meeting with the TA (i.e. 20 academic hours).
The material will be presented rigorously during the class for the most important results and practical examples will be discussed. Each subject is organised and presented to allow students to progress incrementally in the development of their knowledge and skills.
Additional material (lecture notes and exercises) is made available before each class activity to participants. Students are solicited to download or print lecture notes to have them at hand during lectures in order to take their notes.
Lectures will be followed by practice sessions and students are expected to attend the classroom sessions, study the different topics, participate in class contributing with questions and comments.
Attendance to both lectures and practice sessions is strongly recommended. The teacher is weekly available to meet students for consultation during office hours.

English

There is no single textbook for the class: during the course detailed information will be given on content and reading material (page numbers, selected exercises) with reference to main textbooks on the e-learning page of the course (moodle.unive.it). Students are required to register for the course on the Moodle platform.

Accessibility, Disability and Inclusion
Accommodation and support services for students with disabilities and students with specific learning impairments
Ca' Foscari abides by Italian Law (Law 17/1999; Law 170/2010) regarding support services and accommodation available to students with disabilities. This includes students with mobility, visual, hearing and other disabilities (Law 17/1999), and specific learning impairments (Law 170/2010). If you have a disability or impairment that requires accommodations (i.e., alternate testing, readers, note takers or interpreters) please contact the Disability and Accessibility Offices in Student Services: disabilita@unive.it.

written