Mathematics II (TI082)
2015-2016
Tinbergen Institute

### Aim

Mathematics II studies all optimization methods. The course intends to teach the students what are the standard methods of optimization (static/dynamic, discrete/continuous, deterministic/stochastic) and how to use these.

### Contents

This course starts with a brief review of finite dimensional optimization: necessary and sufficient conditions of optimization (stationary, Lagrange, KKT), existence of solutions (Weierstrass) and sensitivity analysis (shadow price). Then indefinite dimensional optimization is discussed: the three methods of dynamic optimization: 1) Calculus of Variation (Euler-Lagrange, transversality), 2) Optimal Control (Pontryagin?s Maximum Principle, Hamiltonians), 3) Dynamic Programming (Bellman). Finally an introduction is given to stochastic dynamic optimization and to dynamic and stochastic games.

### Literature

For the lectures we will mainly use the lecture notes, the book serves as a background reference, and also contains the exercises.

### Examination

We will follow the usual conventions for TI core courses, i.e. there will be a written closed book exam. The exercises at the exam will be at the level of the homework and tutorial sessions, but may also contain some theory. You don't have to know all proofs by heart, but at least the gist of them. Important theorems and definitions you are required to know. The written exam is on October 21, 2015, at 13:30.

### People

Lectures by Peter Spreij, teaching assistance by Huyen Nguyen and Lingwei Kong.

### Schedule

Fall semester, 1st half, on different days in the week. Here is the list of lectures.
1. Wed 2 September, 11:00 - 13:45
2. Fri 11 September, 11:00 - 13:45
3. Fri 18 September, 11:00 - 13:45
4. Wed 23 September, 11:00 - 13:45
5. Wed 30 September, 11:00 - 13:45
6. Wed 7 October, 11:00 - 13:45
7. Wed 14 October, 11:00 - 13:45
For the tutorials, see blackboard for up to date information.

### Location

Tinbergen Institute Amsterdam, Gustav Mahlerplein 117, 1082 MS Amsterdam

 1 Class: Most of Lecture notes Section 1. Tutorial: Homework: Lecture notes exercises Section 1: 1, 2, 4, 8, 9; Textbook exercises D1.4, D1.5, D2.1, D3.2, D3.4 2 Class: Most of Lecture notes Section 2. Plan for the lecture includes: more details on Separation theorem, a little more on calculus for subdifferentials, more details on the proof of the KKT theorem (part 1), examples (numerical illustration) in less detail. Tutorial: Lecture notes exercises Section 2: 6, 9 Homework: Lecture notes exercises Section 2: 1, 2, 3, 5, 8; Textbook exercises D4.2, D4.5, 1.6.6, 1.6.14, 1.6.33 3 Class: (Perhaps quick review of John's theorem, Section 2.4) Most of Section 3, emphasis on Euler-Lagrange and Euler equations; brief mentioning of Pontryagin's maximum principle (connection with Euler-Lagrange); discrete time problem with the inclusion of terminal condition $f(N,X_N)$ in Section 3.4, mentioning of the backward recursion of the $\hat{p}_k$ in Section 3.5. Tutorial: See blackboard Homework: See blackboard 4 Class: Most of Section 4, infinite horizon problems mainly next week. Tutorial: See blackboard Homework: See blackboard 5 Class: Remainder of Section 4 (Bellman equation for infinite horizon), first half of Section 5 (abstract theory). Tutorial: See blackboard Homework: See blackboard 6 Class: Second half of Section 5. Tutorial: See blackboard Homework: See blackboard 7 Class: Tutorial: Homework: