Postgraduate Course: Fundamentals of Optimization (MATH11111)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 11 (Postgraduate)
||Availability||Available to all students
|Summary||Classification of optimization problems; Convexity in optimization; Linear programming: Model formulation and assumptions; Graphical solution; Simplex method; Duality theory; Dual simplex method; Sensitivity analysis; Large-scale linear programming; Unconstrained nonlinear optimization; Optimality conditions.
This course is designed to expose students to different types of optimization problems and to introduce appropriate solution approaches for each type. The role of convexity in optimization is emphasised. The course provides an in-depth treatment of linear programming and solving linear programming problems using the simplex method. The students will be exposed to the theoretical foundations of linear programming problems. The role of duality and sensitivity analysis for linear programming problems are examined. Alternative solution approaches for large-scale linear programming are discussed. The course gives a brief introduction to nonlinear optimization and introduces a few basic algorithms for unconstrained optimization. A tentative list of course topics is as follows: -Introduction, taxonomy of optimization problems, basic examples -Convex sets, convex functions, role of convexity in optimization-Introduction to linear programming, graphical solution, standard form linearprogramming, vertices, simplex method in tableau form -Two-phase simplex method, infeasible and unbounded LPs -Finite convergence of the simplex method -Duality theory -Dual simplex method -Sensitivity analysis, economic interpretation of the dual problem-Large-scale linear programming, column generation, cutting plane methods -Introduction to nonlinear optimization, optimality conditions -Basic algorithms for unconstrained optimization
Entry Requirements (not applicable to Visiting Students)
|| Students MUST have passed:
||Other requirements|| None
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2023/24, Not available to visiting students (SS1)
|Course Start Date
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 12,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||4 assessments (10% each, to be tentatively given out during Weeks 3, 5, 7, and 9) ¿each assessment will have a mix of STACK exercises and open-ended problems. STACK exercises will provide instant individual feedback to students. Open-ended problems will be marked by the course team. The first two learning outcomes will be assessed in each assessment as well as the written exam. The third learning outcome is expected to be assessed in the third and fourth assessments as the topic will be covered in the second half of the semester.
||Before each assessment, a problem set will be announced. Each problem set will have the same format as the following assessment, i.e., a combination of STACK exercises and open-ended problems. STACK exercises will provide instant individual feedback to students. Open-ended problems will be discussed during the workshop in the following week, virtual office hours,and possibly in the discussion forums.
||Hours & Minutes
|Main Exam Diet S1 (December)||Fundamentals of Optimization||2:00|
On completion of this course, the student will be able to:
- Identify different types of optimization problems, and be able to connect these with the available methods for their solution.
- Apply appropriate optimization techniques to solve small optimization problems by hand.
- Discuss and interpret the sensitivity of a solution of an optimization problem to changes in the parameter values of the problem.
|-Introduction to Linear Optimization, Dimitris Bertsimas and John N. Tsitsiklis, Athena Scientific, Dynamic Ideas, LLC, Belmont, Massachusetts, 1997, ISBN: 1886529191 -Linear Programming: Foundations and Extensions, Robert J. Vanderbei; Fred Hillier (Editor); Robert J. Vanderbei (Editor), Springer US, Boston, Massachusetts, 2008, Third Edition, International Series in Operations Research & Management Science, ISBN: 0387743871 -Linear Programming and Network Flows, Mokhtar S. Bazaraa, John J. Jarvis, Hanif D. Sherali, Hoboken, N.J, John Wiley & Sons, 2010, Fourth edition, ISBN: 0471485993-Linear and Nonlinear Programming, David G. Luenberger, Yinyu Ye, Springer US, New York, NY, 2008, Third Edition, International Series in Operations Research & Management Science, ISBN: 0387745025|
|Course organiser||Prof Miguel Anjos
|Course secretary||Miss Gemma Aitchison
Tel: (0131 6)50 9268