Unit of study_

# MATH3971: Convex Analysis and Optimal Control (Adv)

## Overview

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimistion problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straight forward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

### Unit details and rules

Unit code MATH3971 Mathematics and Statistics Academic Operations 6 MATH4071 A mark of 65 or above in 12cp from (MATH2XXX or STAT2XXX or DATA2X02) None MATH2X21 and MATH2X23 and STAT2X11 Yes

### Teaching staff

Coordinator Ben Goldys, beniamin.goldys@sydney.edu.au Ben Goldys

## Assessment

Type Description Weight Due Length
Final exam Final take-home exam
Take home written exam - Week 17
70% - 3 hours
Outcomes assessed:
Assignment Assignment 1
Written work
15% Week 07 Two weeks
Outcomes assessed:
Assignment Assignment 2
Written work
15% Week 12 Two weeks
Outcomes assessed:

### Assessment summary

Detailed information for each assessment can be found on Canvas.

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

The Current Student website  provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Introduction: motivating examples and mathematical formulation of control problems Lecture and tutorial (4 hr)
Week 02 Convex sets, convex cones and convex functions. Static optimisation problems. Legendre-Fenchel transform and duality Lecture and tutorial (4 hr)
Week 03 Optimisation of functions with constraints and KKT conditions Lecture and tutorial (4 hr)
Week 04 Optimal control of ODEs: controllability, observability, stabilisation Lecture and tutorial (4 hr)
Week 05 Optimal control of ODEs: Pontriagin maximum principle Lecture and tutorial (4 hr)
Week 06 Optimal control of ODEs: Dynamic Programming and the Hamilton-Jacobi- Bellman equation Lecture and tutorial (4 hr)
Week 07 Martingales, optional stopping theorem Lecture and tutorial (4 hr)
Week 08 Controlled stochastic systems, Dynamic programming Principle and martingale char- acterisation of the optimal payoff Lecture and tutorial (4 hr)
Week 09 Hamilton-Jacobi-Bellman equation and optimal feedback controls Lecture and tutorial (4 hr)
Week 10 Optimal stopping Lecture and tutorial (4 hr)
Week 11 Kalman filter and Linear-Quadratic Regulator with partial observation for stochastic differential equations Lecture and tutorial (4 hr)
Week 12 One-period games, saddle points and Nash equilibria Lecture and tutorial (4 hr)
Week 13 Dynamic games Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. analyse static optimisation problems with constraints
• LO2. formulate deterministic and stochastic dynamic optimisation problems, that arise in scientific and engineering applications, as mathematical problems
• LO3. understand the importance of convexity for optimisation problems, and use convexity to determine whether a solution to a given problem exists and is unique
• LO4. check if a certain controlled system is controllable, observable, stabilisable
• LO5. apply the Maximum Principle in order to solve real world control problems
• LO6. formulate the Hamilton-Jacobi-Bellman equation for solution of dynamic optimisation problems and solve them in special cases
• LO7. explain the derivations of key theoretical results and discuss the role of mathematical assumptions in these derivations
• LO8. use game theory to formulate optimisation problems with many competing players
• LO9. identify important solvable classes of optimisation problems arising in finance, economics, engineering and insurance and provide solutions.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered.

### Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

• No eating or drinking is allowed in any laboratory under any circumstances
• A laboratory coat and closed-toe shoes are mandatory