Optimization: Theory, Algorithms, and Applications

[203175] burke在这项研究中，我们开发了理论和数值工具来理解和利用谱函数的变分行为。简而言之，谱函数是实数或复值矩阵的谱到实数的映射。两个例子是光谱横坐标（光谱的最大实部）和光谱半径（光谱的最大模量）。谱横坐标和谱半径分别在理解连续和离散动力系统的渐近行为中起着重要作用。由于这个原因，理解它们的变化行为将极大地影响许多应用领域。一般来说，谱函数有许多特征，使它们难以分析。其中最重要的是，它们通常是不可微的，实际上是非李普希茨的。这些函数可能表现得很差，尤其是在需要优化的区域。因此，尽管这些函数在数学、科学和工程中有着经典的历史，但对它们的变分行为的深入了解已被证明是难以捉摸的。由于这个原因，需要结合传统技术的新的分析工具。在这项研究中，我们将变分分析的现代技术与普塞牛顿级数、半代数集、共形映射和多项式几何的经典方法结合在一起。事实证明，这些工具在揭示这类非常重要的函数方面取得了惊人的成功。在优化理论和实践中，人们试图最小化或最大化性能度量，但受制于如何调整性能度量的限制。优化基本上是一个跨学科的研究领域，对广泛的学术、工业和政府研究活动产生重大影响。优化研究需要理论的进步、数值求解方法的发展和应用的坚实基础。本提案中概述的特定研究侧重于与随时间演化的系统稳定性密切相关的优化问题。特别是，它影响建筑物和飞机等结构的设计，这些结构受到地震或风切变等环境因素的时间变形的影响。在这种情况下，潜在的优化问题是使结构在潜在的恶劣环境中尽可能稳定，同时满足某些设计限制，如重量、尺寸和成本。本研究的最大困难在于，所考虑的性能度量，如稳定性，不会随着潜在参数的变化而平稳地变化。因此，从根本上说，需要新的分析方法来理解这些绩效指标的变分行为。事实上，这种分析所需的数学工具直到最近才发展起来。在这项研究中，我们打算在这类问题的分析方面取得重大进展，并开发一系列可用于解决这些问题的数值方法。

0203175Burke 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.