**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 2004). --- 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*.