Convex Optimization for Engineering Analysis and Design

摘要这项研究与课程开发（CRCD）结合的提案的重点是凸优化应用于工程分析和设计。其基本思想是，工程中出现的许多分析和设计问题都可以用凸优化问题来表示，凸优化问题是一种特殊形式的数学问题。虽然这些问题看起来非常困难，但它们可以通过计算机非常有效地解决，使用最近开发的利用凸性的方法。这一思想最近被成功地应用于自动控制系统领域的几个问题，产生了新的理论进展以及实际有用的计算机辅助设计工具的发展。当然，并非所有工程中出现的问题都是凸问题，一些为更一般的数值优化问题开发的方法也可以用于解决凸问题。但是，新的内点方法比一般方法在效率上有巨大的提高潜力，这使得对凸问题的关注是值得的。研究部分包括开发新的和改进的算法来解决工程中出现的凸优化问题。该项目专注于能够利用问题中潜在的工程结构的方法，从而有效地解决甚至大规模的问题。该研究不仅涉及算法的开发和分析，还将涉及对实际问题进行广泛的数值实验。初步实验表明，在工作站几分钟内解决涉及数千个变量和数万个约束条件的问题是可能的。课程开发部分包括创建一套完整的教材，一套传统的教科书，以及通过万维网提供的HTML超文本材料，用于一门关于凸优化与工程应用的新课程。本课程面向所有工程领域的大四本科生和一年级研究生，重点培养在工程应用中识别和利用凸性所需的背景、经验和技能。我们的目标是，这门课程最终应该被每一所拥有高级工程专业的大学采用。最终目标是使凸优化和最近发展的求解凸问题的内点技术进入数值数学的主流，并使我们的主要观点进入工程实践。基于凸优化的工程问题解决方案的实际意义在于为工程问题开发强大的计算机辅助设计和分析工具提供了可能。这些CAD工具的开发和培训学生开发这些CAD工具是对美国整体经济竞争力的明显而实质性的贡献。＊＊＊

9420565 Boyd ABSTRACT The focus of this combined research-curriculum development (CRCD) proposal is convex optimization applied to engineering analysis and design. The basic idea is that many analysis and design problems arising in engineering can be expressed in terms of convex optimization problems, which are mathematical problems of a special form. While these problems can appear very difficult, they can be solved very efficiently by computer, using recently developed methods that exploit convexity. This idea has recently been applied successfully to several problems in the field of automatic control systems, resulting in new theoretical advances as well as the development of practically useful computer-aided design tools. Of course, not all problems arising in engineering are convex, and some methods developed for more general numerical optimization problems can be used to solve convex problems. But the extraordinary potential increase in efficiency of the new interior-point methods over more general methods makes the concentration on convex problems worthwhile. The research component involves the development of new and improved algorithms for the solution of the convex optimization problems that arise in engineering. The project concentrates on methods that can exploit the underlying engineering structure in the problem, and hence solve even large-scale problems efficiently. The research will involve not only the development and analysis of algorithms but also extensive numerical experimentation on real problems. Preliminary experiments suggest the possibility of solving problems involving thousands of variables and tens of thousands of constraints in minutes on a workstation. The curriculum development component involves the creation of a complete set of teaching materials a conventional textbook as well as HTML hypertext material available via World-Wide Web for a new course on convex optimization with engineering applications. The course is aimed at senior under graduates and first year graduate students from all fields of engineering, and will concentrate on developing the background, experience, and skills required to recognize and exploit convexity in engineering applications. The goal is that the course should eventually be adopted at every university with an advanced engineering program. The ultimate goal is that convex optimization and the recently developed interior-point techniques for solving convex problems should make their way into the mainstream of numerical mathematics, and our main point into engineering practice. The practical significance of a convex optimization-based solution to an engineering problem is the possibility of developing powerful computer-aided design and analysis tools for the engineering problem. The development of such CAD tools and the training of students to develop such CAD tools is an obvious and substantial contribution to overall U.S. economic competitiveness. ***