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.

Due to the persistency of the current health emergency, it is not possible to establish now the precise teaching method adopted for this course. We can expect that part of the activities will be delivered in distance (through video recordings and other teaching materials uploaded on the Moodle platform or via lectures held on Hangouts Meet). The description provided here below applies to lessons delivered both in presence or in distance.

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 all the material presented in the course on Mathematics (laurea/bachelor degree):
- Single variable functions.
- Derivatives. Single variable optimization.
- Integrals.
- Several variable functions. Optimization with several variables.
- Matrix algebra and linear equations systems. Applications to economics.

The course program includes presentation of definitions and theoretical results, discussion and solution of problems on the following topics:
- Metric, normed spaces. Contractions. Fixed points. Banach's Theorem.
- Unconstrained Optimization.
- Constrained Optimization (equalities, inequalities).
- Concavity/convexity in Optimization.
- FHDP problems. Bellman Principle. SDP problems. Bellman Principle.
- Calculus of variations. Euler Equation.
- Differential equations.
- Calculus of variations. Second order conditions. Terminal conditions.
- Optimal control. The Maximum Principle. Sufficient conditions.
- Discrete time optimization. The Euler equation. The Maximum Principle revisited.

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

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
Kamien M.I., Schwartz N.L. (1991): Dynamic Optimization, The Calculus of Variations and Optimal Control in Economics and Management, second edition. [Chp.1--6]
Ok, E.A. (2007): Real analysis with economic applications. Princeton University Press.
Sundaram R.K. (1999): A First Course in Optimization Theory, Cambridge University Press, Cambridge. [Chp.11]
Sydsaeter K., Hammmond P., Seierstadt A., Strom A. (2005): Further Mathematics for Economic Analysis, Prentice Hall. [Chp.1--3,5--8]

Grading is based on a final written exam (80% ), problem sets (10%) and attendance (10%); a supplementary oral examination is also possible.
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 the mean value of the two partial examinations.
The remaining sessions (from January until September) present a single comprehensive examination.

The written exam lasting 120' contains 6 with open-ended problems (3+3 in the case of the two partial examinations).
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.
All problems carry equal number of marks. In answering the exam no help can be used (books, notes, computers, calculators, web,…), but students are allowed to use an A4-sheet (one side) prepared by them at home.

The precise form of the final written exam will depend on the health situation and on the disposition of the national authority at the time. Detailed instructions will be given by the instructors in time to prepare for the exam.

There will be thirty lectures in 10 weeks of teaching activities.
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.

Depending on the health situation, updates concerning the activities and their scheduling will be made available on the Moodle platform (moodle.unive.it) and on the personal page of the instructor.

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

This subject deals with topics related to the macro-area "Human capital, health, education" and contributes to the achievement of one or more goals of U. N. Agenda for Sustainable Development