APP MTH 3014 - Optimisation III

North Terrace Campus - Semester 1 - 2016

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. There is always, though, an objective: minimise or maximise some function of the variable(s), subject to the constraints. This course will examine nonlinear mathematical formulations, and will concentrate on convex optimisation problems. Many modern optimisation methods in areas such as design of communication networks, finance, etc, 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, in particular, higher-order Newton's Method, steepest descent methods, conjugate gradient methods; Constrained optimisation, including Kuhn-Tucker conditions and the Gradient Projection Method.

  • General Course Information
    Course Details
    Course Code APP MTH 3014
    Course Optimisation III
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 hours per week
    Available for Study Abroad and Exchange Y
    Prerequisites MATHS 1012
    Assumed Knowledge Knowledge of linear programming such as would be obtained from APP MTH 2105 and basic computer programming skills such as would be obtained from COMP SCI 1012, 1101, MECH ENG 1100, 1102, 1103, 1104, 1105, C&ENVENG 1012
    Course Description 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. There is always, though, an objective: minimise or maximise some function of the variable(s), subject to the constraints. This course will examine nonlinear mathematical formulations, and will concentrate on convex optimisation problems. Many modern optimisation methods in areas such as design of communication networks, finance, etc, 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, in particular, higher-order Newton's Method, steepest descent methods, conjugate gradient methods; Constrained optimisation, including Kuhn-Tucker conditions and the Gradient Projection Method.
    Course Staff

    Course Coordinator: Dr Sarthok Sircar

    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
    University Graduate Attributes

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

    University Graduate Attribute Course Learning Outcome(s)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    1,2,3,4
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    1,3,4,5,6
    Teamwork and communication skills
    • developed from, with, and via the SGDE
    • honed through assessment and practice throughout the program of studies
    • encouraged and valued in all aspects of learning
    5,6
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    1,5,6
    Intercultural and ethical competency
    • adept at operating in other cultures
    • comfortable with different nationalities and social contexts
    • Able to determine and contribute to desirable social outcomes
    • demonstrated by study abroad or with an understanding of indigenous knowledges
    5,6
    Self-awareness and emotional intelligence
    • a capacity for self-reflection and a willingness to engage in self-appraisal
    • open to objective and constructive feedback from supervisors and peers
    • able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
    6
  • Learning Resources
    Required Resources
    None.
    Recommended Resources
    Chong and Zak, An Introduction to Optimization (Wiley).
    Online Learning
    All assignments, tutorials, handouts and solutions, where appropriate, will be made available on MyUni as the course progresses.

    Recordings of lectures will also be available on MyUni following each lecture, for those who are unable to attend due to other commitments and for revision purposes.

    Please don't hesitate to email the lecturer should anything be missing.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures, but lectures will be recorded to help with occasional absences and for revision purposes. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
    Workload

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

    Activity Quantity Workload hours
    Lectures 30 90
    Tutorials 6 18
    Assignments 5 48
    Total 156
    Learning Activities Summary
     Lecture Schedule
    Week 1 Single variable optimisation Introduction, Dichotomous and Golden section searches
    Week 2 Fibbonachi and unbounded searches
    Week 3 Quadratic approximation, DSC algorithm
    Week 4 Unconstrained multi-variable optimisation Newtons method, introduction to unconstrained problems
    Week 5 Convexity, theorems for minimality and descent methods
    Week 6 Steepest descent on quadratics
    Week 7 Conjugate gradient method
    Week 8 Constrained convex optimisation Fletcher-Reeves algorithm, introduction to constrained optimisation
    Week 9 Linear constraints, Lagrange multipliers, KKT conditions
    Week 10 Generalisations of KKT conditions, orthogonal projection
    Week 11 Non-convex optimisation Gradient Projection algorthm, introduction to non-convex optimisation.
    Week 12 Simulated Annealing methods, revision.
    Tutorial Schedule
    Week 2 Dichotomous and Golden Section search methods
    Week 4 Quadratic approximation methods
    Week 6 1-D Newton's method and Convex functions
    Week 8 Steepest Descent Methods
    Week 10 Conjugate Gradient Methods
    Week 12 Non-convex methods


  • 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 70% All
    Assignments Formative and summative 30% All
    Assessment Related Requirements
    An aggregate score of at least 50% is required to pass the course.
    Assessment Detail
    Assessment Item Distributed Due Date Weighting
    Assignment 1 Week 2 Week 3 6%
    Assignment 2 Week 4 Week 5 6%
    Assignment 3 Week 6 Week 7 6%
    Assignment 4 Week 8 Week 9 6%
    Assignment 5 Week 10 Week 11 6%
    Submission
    1. Assignments must be submitted to the correct box in the School of Mathematical Sciences on time with a signed assessment cover sheet attached.
    2. Late assignments will not be accepted.
    3. Assignments will be returned within two weeks to provide feedback to students.
    Course Grading

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

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    FNS   Fail No Submission
    F 1-49 Fail
    P 50-64 Pass
    C 65-74 Credit
    D 75-84 Distinction
    HD 85-100 High Distinction
    CN   Continuing
    NFE   No Formal Examination
    RP   Result Pending

    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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