A Survey of Numerical Algorithms for Optimization
Kaj Madsen, Department of Mathematical Modelling, Technical University of Denmark
Monday June 19, 2000, 2:00pm
CRL, Room B102

Abstract:

During the past 40 years an important part of numerical analysis has been the development of methods for minimizing nonlinear functions. A large and elegant theory supporting the methods has been worked out.

Most of the methods are iterative, and they are based on Taylor's formula: At the current iterate x a model of the nonlinear function f is built, based on function and gradient information available at x. This model is valid at some neighborhood of x and forms the basis for finding the next iterate. The most successful class of methods of this type is the Quasi-Newton methods which can now be considered as fully developed.

If the function f has special characteristics (e.g., being a sum of squares or a minimax function) local models have been used which simulate the special structure of f.

In global optimization other types of methods have been more common. Most often they are based on stochastics. Another prominent example of a global optimization strategy based on more general information about f is the interval method. This method even has a built-in result verification. During the years many researchers have been trying to build local models based on information from several points rather than using the Taylor philosophy. So far this has not been generally successful as compared with the Quasi-Newton principle. However, recent developments have shown promising indications in this direction, mostly through strategies devoted to special applications. Examples are surrogate modeling in aircraft design and space mapping techniques for electromagnetic optimization.

Refreshments and snacks will be served.

For a biography of Kaj Madsen please visit his website http://www.imm.dtu.dk/documents/users/km/homepage.html

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