Objectives and competences
Familiarize the students with the process of mathematical modelling of continuous optimization problems.
Develop competent skills of independent application of mathematical methods to the problems from financial optimization, economics, and broader from industry.
Familiarize the students with technological tools that assist solving optimization problems and problems related to mathematical programming.
Content (Syllabus outline)
Mandatory content, that familiarizes the students with fundamentals of operations research and mathematical programs:
• Unconstrained optimization. Newton's method.
• Constrained optimization. Lagrange multipliers. Necessery and sufficient conditions for a constrained local optimum. Dual of a convex program.
• Quadratic programming. Lagrange methods and active set methods. Programs with linear constraints. Zigzagging.
• Nonlinear programming. Penalty and barrier functions. Lagrange-Newton method. Sequential Quadratic Programming.
• Conic programming. Lorentz and semidefinite cone. Conic quadratic programming.
• Semidefinite programming. Applications in combinatorial optimization.
• Interior point methods for linear and convex programming. Existence of the central path. Primal-dual methods of following the central path.
Within the coursework, the students select a problem - project whose result is coursework report. The problem is related to their future career (practical problems from industry and business, theoretical problems from the areas of optimization, algorithms, modelling). The content of the remaining lectures is selected according to these projects from the following list:
• Price of robustness robust optimization method.
• Portfolio immunization using stohastic programming.
• Stohastic nonlinear programming (discrete and continuous stohastic variables). Decomposition.
• Applications of semidefinite programming: quadratic assignment problem, travelling salesman problem, max cut problem.
• Applications of stohastic programming: Markowitz models of portfolio optimization, multiperiod stohastic planning models.
• Maximum likelihood models, least squares method, parameter fitting for given data.
• Optimization mathematical models from control theory and signal processing.
• Support Vector Machine.
• Other content from the domain of operations research and mathematical programming, related to students' problems.
Within their coursework and exercisces, the students familiarize themselves with software for mathematical modelling, either commercial (Excel, Lindo, Matlab) or freely avaliable open source (SciLab, Neos, R).
Learning and teaching methods
• At the lectures, the students are familiarized with the course content. Applying flipped learning approach, they discuss their coursework projects in relation to the material of the course.
• At the tutorials, the student deepen their understanding of the material by working on an extensive problem related to their future career. They are organized in larger groups who research the choosen problem guided by methodologies of problem-based learning. Within solving the problem, they experience all the stages from requirements and data gathering, model development, selecting and adapting technological solutions to discussing various aspects of implementation of the results.
Intended learning outcomes - knowledge and understanding
• To be able to understand advanced principles of mathematical programming.
• To deepen the knowledge of modern numerical methods for solving mathematical programs.
• To deepen the knowledge of details of Markowitz models and other advanced applications of mathematical programming, financial optimization and wider.
Intended learning outcomes - transferable/key skills and other attributes
• Direct applications in finacial mathematics, economy, business, engineering, physics, and numerous other social and natural sciences.
• Competent mastering of the process of mathematical modelling and applications of its techniques in problems from financial optimization, economics, and wider.
• Preparation of a detailed technical report or focused report paper describing a mathematical model of a specific mathematical problem the students encounter while investigation their possible future careers.
Readings
R. Rardin. Optimization in Operations Research. Prentice Hall, Inc., Upper Saddle River, New Jersey, 2000.
J. Curwin, R. Slater. Quantitive Methods for Business Decisions. Third Edition. Chapman & Hall, London, 1991.
S. A. Zenios, Financial Optimization. Cambridge University Press, Cambridge, 1993.
R. Fletcher, Practical Methods of Optimization. Second Edition. Wiley, Chichester, 2001.
A. Ben-Tal, A. Nemirowski: Lectures on modern convex optimization. H. Milton Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, 2012.
C. Huang, R. H. Litzenberger. Foundations for Finacial Economics. Prentice Hall, Inc., Upper Saddle River, New Jersey, 1988.
P. Kall, S. W. Wallace. Stochastic Programming. Wiley, Chichester, 1994.
L. Neralić, Uvod u matematičko programiranje 1. Udžbenici Sveučilišta u Zagrebu, Zagreb, 2001.
R. Rardin. Optimization in Operations Research. Prentice Hall, Inc., Upper Saddle River, New Jersey, 2000.
J. Renegar. A Mathematical View of Interior-Point Methods in Convex Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia, 2001.
S. A. Zenios, Financial Optimization. Cambridge University Press, Cambridge, 1993.
Prerequisits
Knowledge of simple algorithms.
Knowledge of basic linear algebra and calculus.
Additional information on implementation and assessment Coursework report
Oral exam
Each of the mentioned commitments must be assessed with a passing grade.
Passing grade of the seminar exercise is required for taking the exam.
