Description and Aims:
The course intends to provide the students a thorough understanding of methods for unconstrained and constrained non-linear programming as well as modern methods of global
optimization by combining recent theory with concrete practical 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 following a new text book reflecting advances in the field in the last 15 years along with adequate software.
Application to problems arising in modeling of various fields of computational mathematics, atmospheric and oceanographic sciences, variational data assimilation, finance 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.
The course is intended primarily to graduate students and senior undergraduate students with some background in linear algebra, and programming skills in a high-level language as
well as familiarity with one of the operating computer systems at FSU.
- Linear and Nonlinear Programming, by Stephen G. Nash and Ariela Sofer, McGraw-Hill, 1996, 782pp.
- Optimization Software Guide, by Jorge J. More and Stephen J. Wright, SIAM Frontiers in Applied Mathematics, Vol. 14, 1993, 154pp.
In the last few years the topic of numerical optimization has become the focal point for applications in fields as diverse as atmospheric sciences, in particular variational data assimilation, oceanography, engineering, finance 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 methods of unconstrained minimization. Then we introduce low-storage methods for large-scale unconstrained minimization as well as nonlinear least-squares data fitting. Each method will be computationally illustrated with adequate software and examples as well as hands on experience.
In the second part of this course we start with 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.