Breadcrumb

Stochastic Optimisation (MATH M6005)

Academic Year:

Contents of this document:

Administrative Information

  1. Unit number and title: MATH M6005 Stochastic Optimisation
  2. Level: M/7
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 07/08
  6. Unit Organiser: Vladislav Tadic
  7. Lecturer: Vladislav Tadic
  8. Teaching block: 2
  9. Prerequisites: MATH11300 Probability 1 and MATH 11400 Statistics 1, while MATH21400 Applied Probability 2 is desirable, but not essential.

Unit aims

The underlying aim is to use a combination of models, techniques and theory from stochastic control and equilibrium selection to determine behaviour that is optimal with regard to some given reward structure.

General Description of the Unit

Stochastic optimisation covers a broad framework of problems at the interface of applied probability and optimisation. The main focus of this unit is on Markov decision processes and game theory. Markov decision processes describe a class of single decision-maker optimisation problems that arise when applied probability models (eg Markov chains) are extended to allow for action-dependent transition distributions and associated rewards. Game theory problems are more complex in that they involve two or more decision makers (players), so the optimal action for each player will depend on the actions of other players. Here, we focus on Nash equilibria - strategies that are conditionally optimal in the sense that a player can not do do better by changing their own strategy while other players stay with their current strategy

Each module covers an area of statistics and applied probability relevant to the research and other interests of members of academic staff. Details are given in the Syllabus section below.

Relation to Other Units

This unit is a first course  on stochastic optimisation.

Teaching Methods

Lectures, supported by problem and solution sheets.

Learning Objectives

Students who successfully complete this unit should be able to:

  • recognise and construct appropriate formal Markov decision process (MDP) models and game theoretic models from informal problem descriptions;
  • construct appropriate optimality equations for optimisation problems;
  • understand and use appropriate computational techniques (including dynamic programming and policy and value iteration) to solve finite horizon, and infinite horizon discounted and average cost MDPs;
  • understand the concept of a Nash equilibrium and an evolutionarily stable stategy;
  • compute equilibrium policies for standard and simple non-standard games.

Assessment Methods

The assessment mark for Stochastic Optimisation is calculated from a 1½-hour written examination in APRIL consisting of THREE questions. A candidate's best TWO answers will be used for assessment.
Calculators of an approved type (non-programmable, no text facility) are allowed, i.e., only calculators carrying 'Faculty of Science approved sticker' will be allowed in the examination room.

Award of Credit Points

Credit points will be awarded if a student passes the unit; i.e. attains a final mark of 50 or more.

Transferable Skills

In addition to the general skills associated with other mathematical units, students will also have the opportunity to gain practice in the following: report writing, oral presentations, use of information resources, use of initiative in learning material in other than that provided by the lectures themselves, time management, general IT skills and word-processing.

Texts

1. M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 2005.
2. D. P. Bertsekas, Dynamic Programming and Optimal Control, vol. 1 and 2, 2nd edition, Athena Scientific, 2005.
3. P. Whittle, Optimal Control: Basics and Beyond, Wiley, 1996.
4. R. Gibbons, A Primer in Game Theory, Prentice-Hall, 1992.
5. A. I. Houston and J. M. McNamara, Models of Adaptive Behaviour, Cambridge University Press, 1999.
6. J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982.

Syllabus

1. Markov decision problems (Markov controlled processes, reward/cost structure, optimal strategies).
2. Methods for Markov decision problems (linear programming, policy iteration, value iteration).
3. Static Games (Nash equilibrium, dominance, classification).
4. Population Games (Evolutionary game theory, evolutionary stable strategies, Nash equilibrium).