Optimization: Theory, Algorithms, and Applications

University Of Washington, Seattle WA

Investigators

Abstract

0203175 Burke In this research we develop theoretical and numerical tools for understanding and exploiting the variational behavior of spectral functions. Briefly, spectral functions are mappings of the spectrum of real or complex valued matrices to the real numbers. Two examples are the spectral abscissa (the maximum real part of the spectrum) and the spectral radius (the maximum modulus of the spectrum). The spectral abscissa and the spectral radius play an important role in understanding the asymptotic behavior of continuous and discrete dynamical systems, respectively. For this reason, understanding their variational behavior will greatly impact a number of application areas. In general, spectral functions have a number of features that make them difficult to analyze. Foremost among these is that they are typically nondifferentiable, indeed non-Lipschitzian. These functions can be very poorly behaved especially in regions of interest for optimization. Thus, even though these functions have a classical history in mathematics, science, and engineering, intimate knowledge of their variational behavior has proven elusive. For this reason, new tools of analysis in conjunction with classical techniques are required. In this research we bring together the modern techniques of variational analysis and classical methods of Puiseux-Newton series, semi- algebraic sets, conformal mappings, and the geometry of polynomials. These tools have proven to be phenomenally successful in shedding new light on this very important class of functions. In optimization theory and practice one tries to either minimize or maximize a performance measure subject to limitations on how the performance measure can be adjusted. Optimization is a fundamentally interdisciplinary area of research having a significant impact on a wide range of academic, industrial, and government research activities. Research in optimization requires theoretical advances, the development of numerical solution methods, and a firm grounding in applications. The particular research outlined in this proposal focuses on optimization problems that are closely related to the stability properties of systems that evolve with time. In particular, it impacts the design of structures such as buildings and aircraft that are subject to temporal deformations from environmental factors such as an earthquake or a wind-shear. The underlying optimization problem in this context is to make the structure as stable as possible in a potentially hostile environment while satisfying certain design limitations on such things as weight, size, and cost. The great difficulty in this research is that the performance measures under consideration, such as stability, do not vary in a smooth manner as the underlying parameters vary. Consequently fundamentally new methods of analysis are required to understand the variational behavior of these performance measures. Indeed, the mathematical tools necessary for this kind of analysis have only very recently been developed. In this research we intend to make significant inroads into the analysis of these kinds of problems and to develop a range of numerical methods that can be used to solve them.