FRG: Collaborative Research: Semidefinite optimization and convex algebraic geometry

本提案的目标是发展实代数几何凸集研究的数学基础和相关的计算方法。这项工作需要结合优化、分析、代数和组合学的思想和数学工具。该计划不仅会带来理论见解，还会带来新的算法和软件，使数学、工程等领域的新应用成为可能。这项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将集中于发展一个全面的理论和实用的新算法，为由多项式不等式定义的凸集。具体的问题和技术包括实代数变量凸壳的半定描述，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。数学中的许多领域，以及工程、金融和科学中的应用，都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是如何从多项式方程的代数描述中产生的，以及相应的计算含义。其中一个主要动机是在优化上下文中应用这些结果的可能性。提出的研究将有助于现有的知识，无论是在代数几何技术，以及在数学优化。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）之间创造协同效应。这个项目的成功完成将有助于求解多项式系统的有效和可靠的计算工具的可用性，这具有明确的技术和经济利益。该提案的其他关键特征包括与课程开发、本科生研究项目、研究生和博士后培训以及用于计算优化的新软件工具的开发相结合。

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.