FRG: Collaborative Research: Semidefinite optimization and convex algebraic geometry

这项建议的目的是为研究实代数几何中的凸集发展数学基础和相关的计算方法。这项工作需要将最优化、分析、代数和组合学的思想和数学工具结合在一起。拟议的计划不仅将带来理论上的见解，还将带来新的算法和软件，使其能够在数学、工程和其他领域实现新的应用。这项工作分为五个主要部分：半定规划与平方和、凸半代数集、稀疏性与图形结构、数值多项式优化及其应用、参数的变形与变化。PI将专注于发展由多项式不等式定义的凸集的全面理论和实用的新算法。具体的问题和技巧包括实代数簇的凸壳的半定描述的表示，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。数学中的许多领域，以及在工程、金融和科学中的应用，都需要对凸集有深入的了解。这是一类几何形状，有几种不同但互补的解释。这个项目的目标是更好地理解这些几何性质是如何从它们以多项式方程的代数描述中出现的，以及相应的计算含义。主要动机之一是将这些结果应用于优化的可能性。拟议的研究将对现有的代数几何技术和数学优化方面的知识做出贡献。它将在应用数学的不同分支及其工程和科学应用(例如，在计算生物学和统计建模方面)之间产生协同效应。这个项目的成功完成应该有助于获得有效和可靠的计算工具来求解多项式系统，这些系统具有明显的技术和经济利益。这项建议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以及开发新的计算优化软件工具。

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.