## APP MTH 4114 - Optimisation - Honours

### North Terrace Campus - Semester 1 - 2023

Most problems in life are optimisation problems: what is the best design for a racing kayak? How do you get the best return on your investments? What is the best use of your time in swot vac? What is the shortest route across town for an emergency vehicle? What are the optimal release rates from a dam for environmental flows in a river? Mathematical formulations of such optimisation problems might contain one or many independent variables. There may or may not be constraints on those variables. However, there is always an objective: minimise or maximise some function of the variable(s) subject to the constraints. This course examines nonlinear mathematical formulations, and concentrates on convex optimisation problems. Many modern optimisation methods in areas such as design of communication networks and finance rely on the classical underpinnings covered in this course. Topics covered are: One-dimensional (line) searches, direct methods, polynomial approximation, methods for differentiable functions; Theory of convex and nonconvex functions relevant to optimisation; Multivariable unconstrained optimisation, quasi-Newton Method, steepest descent methods, conjugate gradient methods; Constrained optimisation, including Karush-Kuhn-Tucker conditions and the Gradient Projection Method; Heuristics for non-convex problems, genetic algorithms.

• General Course Information
##### Course Details
Course Code APP MTH 4114 Optimisation - Honours Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week Y MATHS 1012 Basic computer programming skills such as would be obtained from ENG 1002, ENG 1003, COMP SCI 1012, COMP SCI 1101, MECH ENG 1100, MECH ENG 1102, MECH ENG 1103, MECH ENG 1104, MECH ENG 1105, C&ENVENG 1012 Honours students only Most problems in life are optimisation problems: what is the best design for a racing kayak? How do you get the best return on your investments? What is the best use of your time in swot vac? What is the shortest route across town for an emergency vehicle? What are the optimal release rates from a dam for environmental flows in a river? Mathematical formulations of such optimisation problems might contain one or many independent variables. There may or may not be constraints on those variables. However, there is always an objective: minimise or maximise some function of the variable(s) subject to the constraints. This course examines nonlinear mathematical formulations, and concentrates on convex optimisation problems. Many modern optimisation methods in areas such as design of communication networks and finance rely on the classical underpinnings covered in this course. Topics covered are: One-dimensional (line) searches, direct methods, polynomial approximation, methods for differentiable functions; Theory of convex and nonconvex functions relevant to optimisation; Multivariable unconstrained optimisation, quasi-Newton Method, steepest descent methods, conjugate gradient methods; Constrained optimisation, including Karush-Kuhn-Tucker conditions and the Gradient Projection Method; Heuristics for non-convex problems, genetic algorithms.
##### Course Staff

Course Coordinator: Professor Lewis Mitchell

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
1. Understand the complexities of, and techniques for solving, nonlinear optimisation problems.
2. Apply suitable algorithms to one- or multi-dimensional optimisation problems.
3. Understand the theoretical framework underlying the techniques presented in class.
4. Implement computer code for the algorithms as studied in class and critically analyse and interpret the results.
5. Demonstrate skills in communicating mathematics orally and in writing.
6. Demonstrate the ability to work in a group applying the theory developed in this course.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

1,2,3,4

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

1,3,4,5,6

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

5,6

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

1,5,6

Attribute 5: Intercultural and ethical competency

Graduates are responsible and effective global citizens whose personal values and practices are consistent with their roles as responsible members of society.

5,6

Attribute 8: Self-awareness and emotional intelligence

Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

6
• Learning Resources
##### Recommended Resources
Edwin K.P. Chong and Stanislaw H. Zak. An Introduction to Optimization. 3rd edition. John Wiley & Sons, 2008. doi: 10.1002/9781118033340
##### Online Learning
All assignments, tutorials, handouts and solutions, where appropriate, will be made available on MyUni as the course progresses.Recordings of lectures will be available on MyUni following each lecture.
• Learning & Teaching Activities
##### Learning & Teaching Modes
The lecturer guides the students through the course material in short topic videos and a weekly in-person/online consultation session. Students are to engage with the material in these sessions and in private study. Interaction with the lecturer and discussion of any difficulties that arise during the course is encouraged. Students are expected to participate in all sessions and online.  Frequent small homework assignments will promote staged active learning.  Fortnightly assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

Videos 30-50 90
Homework TBD 16
Assignments 5 30
Tutorials 5 20
Total 156
##### Learning Activities Summary
1. Single variable optimisation: Introduction, Dichotomous and Golden section searches, unbounded searches, Quadratic approximation, Newton's and secant methods
2. Unconstrained multi-variable optimisation: introduction to unconstrained problems, Levenberg--Marquardt method, convexity, theorems for minimality and descent methods, Steepest descent on quadratics, Conjugate gradient method, Fletcher-Reeves algorithm
3. Constrained convex optimisation: introduction to constrained optimisation, Linear constraints, Lagrange multipliers, KKT conditions, Generalisations of KKT conditions, orthogonal projection, Gradient Projection algorithm
4. Non-convex optimisation: introduction to non-convex optimisation; methods from Genetic Algorithms, Simulated Annealing, Monte Carlo optimisation
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Assessment task Task type Weighting Learning outcomes Examination Summative 40% All Online test Summative 20% All Assignments, regular homework Formative and summative 35% All Tutorial participation Formative 5% All
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail
Assessment itemDistributedDue dateWeighting
Assignment 1 Week 2 Week 3 5%
Assignment 2 Week 4 Week 5 5%
Assignment 3 Week 6 Week 7 5%
Assignment 4 Week 9 Week 10 5%
Assignment 5 Week 11 Week 13 5%
Regular Homework ongoing ongoing 10%
##### Submission
1. All written assignments are to be either submitted to the designated hand in boxes within the School of Mathematical Sciences with a signed cover sheet attached, or submitted as pdf via MyUni.
2. Late assignments will require a request prior to the due date, and a medical certificate or other documentation.
3. Assignments normally have a one week turn-around time for feedback to students.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M11 (Honours Mark Scheme)
Fail A mark between 1-49 F
Third Class A mark between 50-59 3
Second Class Div B A mark between 60-69 2B
Second Class Div A A mark between 70-79 2A
First Class A mark between 80-100 1
Result Pending An interim result RP
Continuing Continuing CN

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
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