MATH4071: Semester 1, 2020

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Units MATH4071 Semester 1 2020 [Normal day]

Unit outline_

MATH4071: Convex Analysis and Optimal Control

Semester 1, 2020 [Normal day] - Camperdown/Darlington, Sydney

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 optimisation 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 straightforward 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

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	[A mark of 65 or above in 12cp of (MATH2XXX or STAT2XXX or DATA2X02)] or [12cp of (MATH3XXX or STAT3XXX)]
Corequisites ?	None
Prohibitions ?	MATH3971
Assumed knowledge ?	MATH2X21 and MATH2X23 and STAT2X11
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Ben Goldys, beniamin.goldys@sydney.edu.au
Lecturer(s)	Ben Goldys, beniamin.goldys@sydney.edu.au

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.

For more information see guide to grades.

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.

Academic integrity

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.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

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
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.

Outcomes
Graduate qualities

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 payers
LO9. identify important solvable classes of optimisation problems arising in finance, economics, engineering and insurance and provide solutions.

Graduate qualities

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

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

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

There were no changes in the course.

Additional information

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
Follow safety instructions in your manual and posted in laboratories
In case of fire, follow instructions posted outside the laboratory door
First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

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