Mathematical methods for Decision Making


Dynamic optimization with applications

Learning Objectives

At the end of the course the student will be able:

  1. to apply basic dynamic optimization techniques if required during research activities;
  2. to understand and implement research results involving more advanced tools of mathematical programming.

Course Content

The wide branches of optimization theory are commonly accepted as a basic tool to study several topics in both economics and management. Resource allocation, portfolio optimization, inventory management are few but meaningful examples of decision making issues that have been addressed by modeling them as optimization problems. To copy with the increasing number of research papers in management science and economic literature that involves advanced mathematical tools, Ph.D. candidates should master both static and dynamic optimization theory.

The aim of the course is to present some major issues of Dynamic Optimization theory in order to make the student familiar with the topics of applied research and the main assumptions used in the model studied in management science, economics and decision science. A background on standard real analysis, topology, linear algebra and uni-variate and muti-variate differential calculus and integral calculus may be necessary, although the level of skills needed can be mastered with short self study.

The outline of the course will be:

  1. Introduction
    • Definition of a dynamic optimization problem and motivating examples
    • Short presentation of background topics
  2. Differential equations
    1. First order differential equations:
      • Linear differential equations;
      • Separable equations;
      • Other differential equations, qualitative solutions.
    2. Higher order differential equations
      • Second order differential equations;
      • General scheme, systems of differential equations.
  3. Calculus of Variations.
  4. Optimal Control theory.

Course Delivery

Individual study is mandatory to master the topics of the course. Lectures will be led by the instructor, presenting main ideas of the topics. However students will be responsible for reviewing the arguments presented in class and practicing the subject.

A set of Assignment will be provided during the classes to help student reviewing the topics discussed and eventually deepen the study.

Students are also encouraged to actively interact with the instructor, not only during classes, to discuss the studying material.

Required Readings

An exhaustive presentation of the topics of the course, as well as the background arguments, can be found in

  1. Hoy et al. Mathematics for Economics, third edition, The MIT Press; Cambridge, USA, 2011.
  2. J. G. Gift, Contributions to the calculus of variations, Journal of Optimization Theory and Applications, January 1987, Volume 52, Issue 1, pp 25-51
  3. Further readings may be suggested by the instructor during classes.

Interested students can also refer to the following books:

  1. C. Chiang, Fundamental methods of Mathematical Economics, McGraw Hill, 1974
  2. C. Chiang, Elements of Dynamic optimization, McGraw Hill, 1992

Course Evaluation

Grade is based on class assignment (40%) and a final oral exam, based on the discussion of three selected papers.

Syllabus

Session 1 Topics·

  • Introduction.
  • Review of some background topics and setting of the problem.
  • Differential equations
Session 2 Topics

  • First order differential equations
  • Second order differential equations.
  • Higher order differential equations.
Session 3 Topics

  • Dynamic optimization: Calculus of variations.
Session 4 Topics

  • Dynamic Optimization: Optimal control 1.
Session 5 Topics

  • Dynamic Optimization: Optimal control 2.

Topics of each session are subject to vary according to teaching needs.