Ionel Navon's Math Page
Ionel Michael NavonProfessor
Ph.D., University of the Witwatersrand, Johannesburg, Republic of South Africa, 1979
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Courses Taught This Semester
Course Prerequisites The course is intended primarily to graduate students and senior undergraduate students with some background in linear algebra, and with basic knowledge of FORTRAN or C as well as familiarity with one of the operating computer systems at FSU. Talented undergraduate students will be accepted on the basis of permission of the instructor. Graduate students in CSIT, Mathematics and in particular Financial Mathematics, Physics, Meteorology and Oceanography, Chemistry, Economics and Engineering are particularly welcome.
COURSE DESCRIPTION and AIMS The course intends to provide the students a thorough understanding of numerical optimization methods for unconstrained and constrained non-linear programming as well as modern methods of global optimization by combining recent theory with concrete practical and computational experience based on analysis and comparison of efficient up-to-date algorithms for solving real life optimization problems and their implementation on supercomputers, taught by an instructor active in research in optimization. The material will be presented in a manner reflecting most recent advances in the field during the last 15 years along with adequate software to illustrate each method.
Application to problems arising in modeling of various aspects of financial mathematics computational mathematics, atmospheric and oceanographic sciences, variational data assimilation and economics, optimal control and engineering applications will be emphasized along with new optimization software and its implementation for multidisciplinary computational science and engineering oriented research projects.
Details In the last few years the topic of numerical optimization has become the focal point for applications in fields as diverse as financial mathematics, atmospheric sciences (in particular variational data assimilation), oceanography, engineering and economics. The desire to solve a problem in an optimal way is so common that optimization models arise in almost every area of application.
The last few decades have witnessed astonishing improvements in computer hardware and these advances have made optimization models a practical tool in business, science and engineering. It is now possible to solve problems with millions of variables. The theory and algorithms that make this possible form a large portion of this course.
In the first part of this course we start by presenting basics of unconstrained minimization followed by the leading algorithms of unconstrained minimization. We then introduce low-storage methods for large-scale unconstrained minimization as well as nonlinear least-squares data fitting.
Each method is computationally illustrated with adequate software from major numerical optimization software libraries collections and examples as well as hands on experience.
This experience is provided by working on theory and problems related to class lectures as well as by using interactive web access to the Network Enabled Optimization Server ( NEOS) at Argonne National Laboratory which is a high speed, socket-based interface for UNIX workstations that provides easy access to all the optimization solvers available with the NEOS Server and which be integrated in the course.
This tool allows users to submit problems to the NEOS Server directly from their local networks. Results are displayed on the screen.
The NEOS guide at http://www.mcs.anl.gov/otc/Guide contains information about most areas of optimization and presents a number of case studies of application of optimization algorithms to real world problems such as portfolio optimization and optimal dieting.
In the second part of this course we study optimality conditions for constrained optimization followed by feasible point methods such as Sequential Quadratic Programming and Reduced Gradient Methods. We then proceed to penalty and barrier methods as well as multiplier based methods such as Augmented Lagrangian method.
Finally we dedicate some attention to global minimization methods dealing in particular with simulated annealing and genetic algorithms and illustrate them with applications. The main sections of the course are prepared with a view of being also presented on the web . The course grades will be based 50% on homework and 50% on a mid-term and final examination.