# Optimization and Control

Course information, a blog, discussion and resources for a course of 16 lectures on Optimization and Control to third year mathematicians at Cambridge in winter 2013.

Until recently my home page linked to content for the 2012 course. Click HERE if you want that.

PAGES      RESOURCES

## Course information

This is a home page of information for Richard Weber's course of 16 lectures to third year Cambridge mathematics students in winter 2013, starting January 17, 2013 (Tue/Thu @ 9am in CMS meeting room 3). This material is provided for students, supervisors (and others) to freely use in connection with this course. A web page for the 2012 course home page is also still available.

## Introduction

This is a course on optimization problems that are posed over time. Importantly we now add dynamic and stochastic elements that were not present in the IB Optimization course, and we consider what happens when some variables are imperfectly observed. Applications of this course are to be found in science, economics and engineering (e.g., "insects as optimizers", "planning for retirement", "finding a parking space", "optimal gambling" and "steering a space craft to the moon").

There is no other course that is an absolute prerequisite, although acquaintance with Probability, Markov chains and Lagrangian multipliers is helpful. However, you should not worry if you did not attend these courses, as there is really nothing you will need to know that cannot be picked up along the way. An important part of the course is to show you a range of applications. The course should appeal to those who liked the type of problems occurring in Optimization IB, Markov Chains IB and Probability IA.

The diagram of the trolley on wheels at the top of this page models the problem that a juggler has when she tries to balance a broom upright in one hand. In this course you will learn how to balance N brooms simultaneously, on top of one another, or perhaps side by side. Can you guess which of these two juggling trick is more difficult?

## Course notes

I aim to make each lecture a self-contained unit on a topic, with notes of four A4 pages.

Here are my starting course notes 2013. Initially, these are identical to the final version of the course notes 2012. But I will probably make some changes as the course proceeds. You can download a copy now, and take another copy once the course ends.

These notes include:

Lectures 1 – 16
Slides used in Lecture 7: Multi-armed bandits and Gittins index
Index

The notes contain hyperlinks, but you may not see these in your browser. You will have to download a copy of the pdf file. If you click on an entry in the table of contents, or on a page number in the Index, you be taken to the appropriate page.

If you think you find a mistake in these notes, please check that you have the most recent copy (as I may have already made a correction.) Otherwise, please let me know and I will incorporate that correction into the notes.

There also exists a nice set of notes for this course by James Norris.

## Course Blog

There is a course blog in which I will make a few comments after each lecture: to emphasize an idea, give a sidebar, correction (!), or answer an interesting question (that perhaps a student sends to me in email).

## Example Sheets

There are 3 examples sheets, each containing 11 questions, in this single file: exsheetoc.pdf. You should receive a supervision on each examples sheet.

This year I am organising these pages so that students can comment or ask questions on the discussion page, and also at the end of each of the blog posts. I will endeavour to answer questions. When you comment, you can do this anonymously if you wish. Just give any name and leave the email address box blank. You can also send me email with questions, suggestions or if you think you have discovered a mistake in the notes.

## Feedback

This year's course will finish on Tuesday 12 March 2013. The written feedback questionnaire has been completed by __ students, and a further __ have completed the electronic form. If you have not yet given feedback you can do so using my equivalent on-line form. This will be sent to my email anonymously. After reading the responses, I will forward them to the Faculty Office.

## Exam questions

Here is a file of all tripos questions in Optimization and Control for 2001-present.

Here are past tripos questions 1995-2006: tripos.pdf

Some of these questions have amusing real-life applications.

## Recommended books

You should be able these books in libraries around Cambridge. Clicking on the title link will show you where the book can be found.

Bertsekas, D. P., Dynamic Programming and Optimal Control, Volumes I and II, Prentice Hall, 3rd edition 2005. (Useful for all parts of the course.) ISBN 1886529086

Hocking, L. M., Optimal Control: An introduction to the theory and applications, Oxford 1991. (Chapters 4-7 are good for Part III of the course.) ISBN 0198596820

Ross, S., Introduction to Stochastic Dynamic Programming. Academic Press, 1995. (Chapters I-V are useful for Part I of the course.) ISBN 0125984219

Whittle, P., Optimization Over Time. Volumes I and II. Wiley, 1982-83.(Chapters 5, 17 and 18 of volume I are useful for Part II of the course. Chapter 7, volume II is good for Part III of the course.) ISBN 0471101206

## Schedules

Dynamic programming
The principle of optimality. The dynamic programming equation for ﬁnite-horizon problems. Interchange arguments. Markov decision processes in discrete time. Inﬁnite-horizon problems: positive, negative and discounted cases. Value interation. Policy improvment algorithm. Stopping problems. Average-cost programming. [6]

LQG systems
Linear dynamics, quadratic costs, Gaussian noise. The Riccati recursion. Controllability. Stabilizability. Inﬁnite-horizon LQ regulation. Observability. Imperfect state observation and the Kalman ﬁlter. Certainty equivalence control. [5]

Continuous-time models
The optimality equation in continuous time. Pontryagin’s maximum principle. Heuristic proof and connection with Lagrangian methods. Transversality conditions. Optimality equations for Markov jump processes and diﬀusion processes. [5]

## Contents

1 Dynamic Programming
1.1 Control as optimization over time
1.2 The principle of optimality
1.3 Example: the shortest path problem
1.4 The optimality equation
1.5 Markov decision processes
2 Examples of Dynamic Programming
2.1 Example: managing spending consumption and savings
2.2 Example: exercising a stock option
2.3 Example: secretary problem
3 Dynamic Programming over the Infinite Horizon
3.1 Discounted costs
3.2 Example: job scheduling
3.3 The operator formulation of the optimality equation
3.4 The infinite-horizon case
3.5 The optimality equation in the infinite-horizon case
3.6 Example: selling an asset
3.7 Example: minimizing flow time on a single machine
4 Positive Programming
4.1 Example: possible lack of an optimal policy.
4.2 Characterization of the optimal policy
4.3 Example: optimal gambling
4.4 Value iteration
4.5 Example: pharmaceutical trials
5 Negative Programming
5.1 Example: a partially observed MDP
5.2 Stationary policies
5.3 Characterization of the optimal policy
5.4 Optimal stopping over a finite horizon
5.5 Example: optimal parking
6 Optimal stopping problems
6.1 Bruss's odds algorithm
6.2 Example: Stopping a random walk
6.3 Optimal stopping over the infinite horizon
6.4 Sequential Probability Ratio Test
6.5 Bandit processes
7 Bandit Processes and the Gittins Index
7.1 Multi-armed bandit problem
7.2 The two-armed bandit
7.3 Gittins index theorem
7.4 Playing golf with many balls
7.5 Example: Weitzman's problem
7.6 *Calculation of the Gittins index*
7.7 *Forward induction policies*
7.8 *Proof of the Gittins index theorem*
8 Average-cost Programming
8.1 Average-cost optimality equation
8.2 Example: admission control at a queue
8.3 Value iteration bounds
8.4 Policy improvement algorithm
9 Continuous-time Markov Decision Processes
9.1 Stochastic scheduling on parallel machines
9.2 Controlled Markov jump processes
9.3 Example: admission control at a queue
10 LQ Regulation
10.1 The LQ regulation problem
10.2 The Riccati recursion
10.3 White noise disturbances
10.4 LQ regulation in continuous-time
10.5 Linearization of nonlinear models
11 Controllability and Observability
11.1 Controllability and Observability
11.2 Controllability
11.3 Controllability in continuous-time
11.4 Example: broom balancing
11.5 Stabilizability
11.6 Example: pendulum
11.7 Example: satellite in a plane orbit
12 Observability and the LQG Model
12.1 Infinite horizon limits
12.2 Observability
12.3 Observability in continuous-time
12.4 Example: satellite in planar orbit
12.5 Imperfect state observation with noise
13 Kalman Filter and Certainty Equivalence
13.1 The Kalman filter
13.2 Certainty equivalence
13.3 The Hamilton-Jacobi-Bellman equation
13.4 Example: LQ regulation
14 Pontryagin's Maximum Principle
14.1 Example: parking a rocket car
14.2 Heuristic derivation of Pontryagin's maximum principle
14.3 Example: parking a rocket car (continued)
14.4 Adjoint variables as Lagrange multipliers
15 Uses of Pontryagin's Maximum Principle
15.1 Transversality conditions
15.2 Example: use of transversality conditions
15.3 Problems in which time appears
15.4 Example: monopolist
15.5 Example: insects as optimizers
15.6 Problems with terminal conditions
15.7 Neoclassical economic growth
16 Controlled Diffusion Processes
16.1 The dynamic programming equation
16.2 Diffusion processes and controlled diffusion processes
16.3 Example: noisy LQ regulation in continuous time
16.4 Example: passage to a stopping set