**Applications:** Control theory, signal analysis, statistics, economics,
diverse industrial problems etc.

**Schedule:** The course gives 4 credits and is offered during the
first half of the autumn (lp 1). It containes 28 lectures, 14 problem solving
sessions, and also two computer laboratory exercises of 2 hours each,
where MATLAB is used to demonstrate algorithms. There is also an individual
programming project that usually takes a couple of days to complete. Written
(occasionally oral) examination.

** Character:**The aim of the course is to give the mathematical ideas
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 Böiers, Lectures on Optimization (Lund 2005).
--- These notes cover the course. Complementary reading: Bazaraa-Sheraly-Shetty,
Nonlinear Programming, Theory and Algorithms, 2nd edition, Wiley 1993.

**Lecturer:** Andrey Ghulchak,
tel. 046/222 8546

Problems in mathematics and its applications very often end up in the minimization or maximization of some function of several variables, possibly constrained. 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*.