Applications: Control theory, signal analysis, statistics, economics, various industrial problems, etc.
Schedule: The course gives 6 credits and is offered during the first half of the autumn (lp 1). It comprises 28 hours of lectures, 14 hours of problem solving sessions, 4 seminars and two computer laboratory exercises of 2 hours each, where MATLAB is used to demonstrate algorithms. There is also an individual programming project which usually takes a couple of days to complete. Written examination on October 16. The programming project should be completed and accepted before February 1. Since usually changes are required, a first version should be submitted several weeks earlier.This year there will be an introductory seminar in the second week of the course discussing the project and reviewing parts of Matlab.
Character:The aim of the course is to give the mathematical ideas behind, and derivations of, the basic general optimization methods because short ''recipes'' cannot be given. Thus there is more emphasis on derivations, different interpretations and proofs than in the compulsory courses in mathematics at LTH, even though the major part of the exercises and the examination is problem solving.
Literature: L-C B÷iers, Mathematical Methods of Optimization (Studentlitteratur 2010) . --- These notes cover
Lecturer: Lars-Christer B÷iers, tel. 046/222 8562
Problems in mathematics and its applications very often end up in the minimization or maximization of some function of several variables, possibly with constraints ---restrictions on which values the variables may take. A common situation is the determination of parameters in a physical model to obtain the best agreement with some set of measured data. Another one is to find an optimal way to transmit information from one point to another in a network.
Differential calculus can often be used to formulate conditions for optimality. This can already be seen in introductory courses in calculus, notably in the theory of Lagrange multipliers. In the present course this is generalized to more complicated situations by the Kuhn-Tucker theory. A fundamental concept here is convexity. Differential calculus is also used in the construction of numerical methods for optimization, together with a good deal of linear algebra. The algorithms are often iterative. The course deals with the basic methods for unconstrained optimization such as Steepest Descent, Newton's Method, Quasi-Newton Methods and the Conjugate Gradient Method. In the presence of constraints, the task of optimization becomes much harder, especially in the case of non-linear constraints. Some general methods and ideas will be presented. In linear programming both the function to be optimized and the constraints are linear. Such problems frequently arise in practice, often in situations involving thousands of variables, and the availability of fast algorithms is of great economic importance. The most important method used here is the simplex method.