FRG: Collaborative Research: Semidefinite optimization and convex algebraic geometry

FRG：协作研究：半定优化和凸代数几何

• 批准号: 0757207
• 负责人: Pablo Parrilo
• 金额: $ 24万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757207&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Semidefinite; optimization

The goal in this proposal is to develop the mathematical foundations andassociated computational methods for the study of convex sets in realalgebraic geometry. This work requires a combination of ideas andmathematical tools from optimization, analysis, algebra and combinatorics.The proposed program will lead not only to theoretical insights, but also tonew algorithms and software that will enable novel applications inmathematics, engineering, and beyond. The work is organized in five mainthrusts: semidefinite programming and sums of squares, convex semi-algebraicsets, sparsity and graphical structure, numerical polynomial optimizationand applications, and deformations and variation of parameters. The PIs willfocus on the development of a comprehensive theory and practical newalgorithms for convex sets defined by polynomial inequalities. Specificproblems and techniques include the formulation of semidefinite descriptionsof convex hulls of real algebraic varieties, determinantal representationsof hyperbolic polynomials, sparse polynomials and their symmetries, tropicalgeometry and homotopy techniques, and geometric programming.Many areas in mathematics, as well as applications in engineering, financeand the sciences, require a thorough understanding of convex sets. This is aclass of geometric shapes, with several different but complementaryinterpretations. The goal in this project is to achieve a betterunderstanding of how these geometric properties emerge from their algebraicdescriptions in terms of polynomial equations, and the correspondingcomputational implications. One of the main motivations is the possibilityof applying these results in the context of optimization. The proposedresearch will contribute to existing knowledge, both in algebraic-geometrictechniques as well as in mathematical optimization. It will create synergiesbetween different branches of applied mathematics, and their engineering andscientific applications (e.g., in computational biology and statisticalmodeling). Successful completion of this project should contribute to theavailability of efficient and reliable computational tools for solvingpolynomial systems, which have clear technological and economic interest.Other key features of this proposal include its integration with curriculumdevelopment, undergraduate research projects, training of graduate studentsand postdocs, and the development of new software tools for computationaloptimization.

该提案的目标是为实代数几何中凸集的研究开发数学基础和相关计算方法。这项工作需要结合优化、分析、代数和组合数学的思想和数学工具。所提出的程序不仅会带来理论见解，还会带来新的算法和软件，从而在数学、工程等领域实现新的应用。该工作分为五个主要方向：半定规划和平方和、凸半代数集、稀疏性和图形结构、数值多项式优化和应用以及参数的变形和变化。 PI 将专注于开发由多项式不等式定义的凸集的综合理论和实用新算法。具体问题和技术包括实代数簇凸包的半定描述、双曲多项式的行列式表示、稀疏多项式及其对称性、热带几何和同伦技术以及几何规划。数学的许多领域以及工程、金融和数学领域的应用科学，需要对凸集有透彻的理解。这是一类几何形状，有几种不同但互补的解释。该项目的目标是更好地理解这些几何性质如何从多项式方程的代数描述中得出，以及相应的计算含义。主要动机之一是可以在优化的背景下应用这些结果。拟议的研究将为代数几何技术以及数学优化方面的现有知识做出贡献。它将在应用数学的不同分支及其工程和科学应用（例如，计算生物学和统计建模）之间产生协同作用。该项目的成功完成应有助于提供高效可靠的计算工具来求解多项式系统，这具有明确的技术和经济利益。该提案的其他主要特点包括与课程开发、本科生研究项目、研究生和博士后培训的结合，以及开发用于计算优化的新软件工具。