37400 Postgraduate Optimisation
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Credit points: 8 cp
Result type: Grade and marks
There are course requisites for this subject. See access conditions.
Anti-requisite(s): 37343 Nonlinear Methods in Quantitative Management
Description
This subject presents a range of concepts and techniques commonly used in solving nonlinear optimisation problems, arising in engineering, computer science, statistics, finance, and economics. Topics presented include Newton's and conjugate direction methods for unconstrained nonlinear programming as well as feasible direction methods, and penalty and barrier methods for constrained nonlinear programming. Another mathematical technique, widely used in practice from production planning and scheduling to personnel rostering and educational timetabling, is linear programming. The corresponding linear programming models often are solved using the interior point methods, which are based on nonlinear programming. The subject provides a brief introduction to the interior point methods.
The theoretical concepts covered are applied to solve real world problems in statistics, finance and economics.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | formulate nonlinear programming models for the problems, considered in this subject; |
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| 2. | communicate and use the definitions and theorems, studied in this subject, including conditions for optimality and conditions for convexity; |
| 3. | solve nonlinear programming problems, using the optimisation algorithms, studied in this subject; |
| 4. | solve nonlinear programming problems using professional software |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Analyse advanced knowledge and critically evaluate the information's source and relevance, with a focus on applications of mathematical methodologies to quantitative finance problem solving. (1.1)
- Apply research to complex real-world problems in the areas of quantitative finance by critically evaluating information and solutions and conducting appropriate approaches to independent research. (2.1)
- Work ethically and confidentially in an organised and collaborative way whilst managing data and applying mathematical and statistical modelling to quantitative finance problems. (3.1)
- Reflect on the value, integrity, and relevance of multiple sources of information to derive creative solutions using technologies to solve quantitative finance problems. (4.1)
- Identify and present complex ideas and justifications using appropriate communication approaches from a variety of methods (oral, written, visual) to communicate with mathematicians, data analysts, scientists, industry, and the general public. (5.1)
Contribution to the development of graduate attributes
This subject also contributes specifically to the development of the following course intended learning outcomes:
Graduate attributes 1.0: Disciplinary knowledge
The definitions and theorems, presented in the subject, provide students with disciplinary knowledge in optimisation, which is one of the major fields of applied mathematics. This theoretical foundation enables the acquisition of professional skills in the form of a number of optimisation algorithms that are widely used in practice. The subject also provides examples of real world problems in finance, engineering and other applications, amenable to the methods of nonlinear programming.
Graduate Attribute 2.0: Research, inquiry and critical thinking
Throughout the semester, you will be given extensive opportunities in the lectures, computer labs and assessment tasks to identify, design and implement multiple approaches to model real world problems and to compare the solutions from those approaches.
Graduate Attribute 3.0: Professional, ethical, and social responsibility
The ability to work effectively and efficiently in a team is a key professional skill. This subject involves multiple tasks where teamwork is required. You will also have the opportunity to develop your skills in using specialist mathematical/statistical/QM Software to implement mathematical approaches to solve problems relevant to industry and public policy development. Professional and ethical responsibilities are emphasised throughout the computational experiments, from data collection, documentation of algorithm development and implementation, to the repeatability of experimental results.
Graduate attribute 4.0: Reflection, Innovation, Creativity
The assessment tasks in this subject and specifically designed learning activities teach students how to work with mathematical literature. In particular, students will be asked to learn some optimisation techniques and applications of nonlinear programming that are not presented in the lectures. This develops the ability for life-long intellectual development.
Graduate attribute 5.0: Communication
Numerous activities in this subject will facilitate the development of communication skills, including all class tests and assignments. Students will be encouraged to do the assignment in groups. The assignment will require a written report.
Teaching and learning strategies
The subject is taught using a combination of lectures, interactive workshops and computer labs according to the topic discussed in that session.
The lectures will involve the presentation, discussion, and exploration of problems, algorithms, theoretical results, or practical applications. Students will be encouraged and are expected to contribute actively to the discussions, where the experience in solving various optimisation problems using the presented solution algorithms will be provided.
The workshops will provide opportunities for students to apply relevant algorithms in practice questions. Students will be presented with nonlinear optimisation problems which can be solved using the algorithms studied in the lectures. Students are expected to attempt workshop questions provided on Canvas during the workshops and will be encouraged to actively participate in the workshop discussions.
In the lab sessions, verbal and written feedback on their progress is provided to the students by the lecturer/tutors from week 1 across the entire semester. During the computer lab sessions, software packages will be demonstrated, and it is expected that students will use these extensively for solving problems and in assignments. Students will collaborate in the lab sessions in small groups around some key focus questions and interact with the whole class and the tutor to receive in-class verbal feedback from tutors and peers.
In order to achieve effective and successful T&L (teaching and learning), it is expected that students actively participate in all T&L activities and submit all assessment tasks. The recommended reading and references are provided as an aid to students’ learning. Canvas is used as the main medium for communication and learning support. Relevant information will be announced on Canvas or sent to the students via email. A student can use only the UTS email account which is registered for this subject. It is the student's responsibility to have the email account in working condition and to check it regularly.
Assessment task 3, “Problem Solving Assignment”, builds on a real-world problem that needs to be approved first by the instructor.
Content (topics)
The subject presents a number of optimisation methods, which are among the most commonly used in practice, together with the theoretical results justifying these methods. These theoretical results are important for successful application of the considered optimisation methods and will further develop your ability to learn new mathematical techniques independently. The subject is comprised of the following topics:
- Nonlinear programming models in finance, engineering and other applications.
- First and second order optimality conditions.
- Convex sets and convex functions.
- Unconstrained optimisation
- Nonlinear constrained optimisation.
- Introduction to the interior point methods.
Assessment
Assessment task 1: Class test 1
| Intent: | This assessment task contributes to the development of the following graduate attributes: 2. research, inquiry and critical thinking (2.1) 5. communication (5.1) |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2 and 3 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1 and 5.1 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 20% |
| Length: | 2.5h in the computer lab |
| Criteria: | 1. Correct application of fundamental principles and concepts of mathematical programming in optimisation analysis |
Assessment task 2: Class test 2
| Intent: | This assessment task contributes to the development of the following graduate attributes: 2. research, inquiry and critical thinking (2.1) 5. communication (5.1) |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2 and 3 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1 and 5.1 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 30% |
| Length: | 2.5h in the computer lab |
| Criteria: | 1. Correct application of fundamental principles and concepts of mathematical programming in optimisation analysis |
Assessment task 3: Problem Solving Assignment
| Intent: | This assessment task contributes to the development of the following graduate attributes: 2.0 Research, inquiry and critical thinking (2.2) 3.0 Professional, ethical, and social responsibility (3.1) 4.0 Reflection, Innovation, Creativity (4.1,4.3) 5.0 Communication (5.1, 5.3) |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1, 4.1 and 5.1 |
| Type: | Report |
| Groupwork: | Group, group and individually assessed |
| Weight: | 50% |
| Criteria: | Group/Individual 30/20 Mathematical correctness of the answer |
Minimum requirements
Students must obtain an overall mark of at least 50 to pass this subject.
References
1. Antoniou A. and Lu W-S., Practical Optimisation Algorithms and Engineering Applications, Springer, 2007.
2. Bartholomew-Biggs M., Nonlinear Optimization with Financial Applications, Kluwer Academic Publisher, 2005.
3. Bazaraa M. S., Sherali H. D., and Shetty C. M., Nonlinear Programming (theory and algorithms), third ed. John Wiley & Sons, Inc., 2006.
4. Belegundu A.D. and Chandrupatla T.R., Optimization Concepts and Applications in Engineering, Prentice Hall, 1999.
5. Bertsekas D. P. Nonlinear programming, Athena Scientific, 1995.
6. Birge J. R. and Louveaux F. Introduction to Stochastic Programming, Springer, 1997.
7. Bonnans J.F., Gilbert J.C., Lemarechal C. and Sagastizabal C.A., Numerical Optimization: theoretical and practical aspects, Springer, 1997.
8. Carvalho P.C.P., de Figueiredo L.H., Gomes J. and Velho L., Mathematical Optimization in Computer Graphics and Vision, Elsevier, 2008.
9. Castillo E., Conejo A.J., Pedregal P., Garcia R. and Alguacil N., Building and solving Mathematical Programming Models in Engineering and Science, John Wiley & Sons, 2002.
10. Chong E.K.P. and Zak S., An Introduction to Optimization, second edition, John Wiley & Sons, 2001.
11. Cornuejols G. and Tutuncu R., Optimization Methods in Finance, Cambridge University Press, 2007.
12. Fletcher R. Practical Methods of Optimization, second edition, John Wiley & Sons, 2000.
13. Jongen H.Th., Meer K. and Triesch E., Optimization Theory, Kluwer Academic Publisher, 2003.
14. Joshi M.C. and Moudgalya K.M., Optimization: theory and practice, Alpha Science International Ltd., 2004.
15. Kall P. and Wallace S.W., Stochastic Programming, Chichester ; Wiley, c1994.
16. Luenberger D. G., Linear and Nonlinear Programming, second edition, Kluwer Academic Publisher, 2003.
17. Miller R.E., Optimisation: foundations and applications, John Wiley & Sons Ltd., 2000.
18. Murty K. G. Operations Research (deterministic optimisation models), Prentice Hall, 1995.
19. Nash S.G. and Sofer A., Linear and Nonlinear Programming. McGraw-Hill International Editions, 1996.
20. Nocedal J. and Wright S.J., Numerical Optimisation, Springer, 1999.
21. Rao S. S. Engineering Optimisation, John Wiley and Sons, Inc., 1996.
22. Rarsin R.L., Optimisation in Operations Research, Prentice Hall, 1998.
23. Ravindran A., Ragsdell K.M. and Reklaitis G.V., Engineering Optimization, Methods and Applications, (second ed.) John Wiley & Sons, 2006.
24. Roos C., Terlaky T. and Vial J.-Ph., Interior Point Methods for Linear Optimization, second edition, Springer, 2006.
25. Venkataraman P., Applied Optimization with MATLAB Programming, John Wiley & Sons, 2009.