Collaborative Research: AF: Small: On the Complexity of Semidefinite and Polynomial Optimization through the Lens of Real Algebraic Geometry

合作研究：AF：小：通过实代数几何的视角探讨半定和多项式优化的复杂性

• 批准号: 2128527
• 负责人: Tamas Terlaky
• 金额: $ 25万
• 依托单位: Lehigh University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-10-01 至 2024-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2128527&HistoricalAwards=false
• 关键词: Collaborative; Research; AF; Small; Complexity

Semidefinite and polynomial optimization (SDO and PO) are topics of great theoretical andpractical interest, with numerous applications in theoretical computer science, control theory,quantum information sciences, and statistics. The steady advances in efficient interior-pointmethods (IPMs) lends credence to the impactful role of SDO, as an emerging computationaltool, in PO and quantum computing. The complexity of SDO and PO is well-known in thebit model of computation: there is no polynomial-time algorithm yet to find an exact optimalsolution of these classes of optimization problems. However, even for an approximate solution,there are pathological instances that IPMs or relaxation hierarchies for PO fail to solve.In view of this challenge, the need to investigate the complexity through a broader spectrumof complexity measures is obvious. Such a novel approach allows for a finer classification ofinstances with high complexity. This project pursues the above goal by addressing severalkey questions on the complexity of SDO and PO, through the lens of real algebraic geometry.The results of this project will enhance understanding of the complexity in SDO and POand have the potential to impact other disciplines, including quantum information sciences,where the emerging area of quantum IPMs with their unique advantages offer unprecedentedintellectual challenges. Due to the multidisciplinary nature of this project, the investigators willtrain graduate students and organize meetings by inviting experts as well as young researchersfor fruitful interaction amongst the optimization and real algebraic geometry communities.The first part of the project focuses on quantitative and algorithmic questions about the complexityof SDO from the perspective of the central path. Since IPMs operate in a neighborhoodof the central path, their efficiency is influenced by the analytic and algebro-geometric propertiesof the central path. The investigators explore several complexity measures based on thedegree, worst-case convergence rate, and geometric curvature of the central path for regularand near to ill-posed instances. By means of these complexity measures, in particular, onecan quantitatively justify the complexity of IPMs on instances whose special structures exhibitfailure of the strict complementarity condition. The second part of the project investigates thecomplexity of PO through error bounds and the topology of the feasible set. Unlike IPMs, themoment/sum of squares approach only deals with a sequence of objective values (rather thansolutions), which may not adequately reflect the progress toward the optimal set. As a result,the current complexity bounds from the moment/sum of squares approach are purely algebraicwith no reliance on the topology/geometry of the feasible set. The investigators will exploretopology based complexity measures which allow for the inclusion of the Betti numbers of thefeasible set and thus enhance PO solvers with more precise complexity estimates. That willrigorously explain why iterative algorithms are more likely to stop at a local optima of a POproblem, as the number of connected components or the holes in the feasible set increases.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

半定和多项式优化（SDO和PO）是理论和实际上都很有意义的问题，在理论计算机科学、控制理论、量子信息科学和统计学中有着广泛的应用。有效的通道点方法（IPM）的稳步发展使人们相信SDO作为一种新兴的计算工具在PO和量子计算中的重要作用。SDO和PO的复杂性在位计算模型中是众所周知的：还没有多项式时间算法来找到这些优化问题的精确最优解。然而，即使是一个近似的解决方案，也有病理情况下，IPM或放松层次的PO无法解决。鉴于这一挑战，需要通过更广泛的复杂性措施调查的复杂性是显而易见的。这种新颖的方法允许对具有高复杂性的情况进行更精细的分类。本项目通过真实的代数几何的透镜，解决SDO和PO复杂性的几个关键问题，从而实现上述目标。本项目的结果将加深对SDO和PO复杂性的理解，并有可能影响其他学科，包括量子信息科学，其中新兴的量子IPM领域以其独特的优势提供了前所未有的智力挑战。由于该项目的多学科性质，研究人员将通过邀请专家和年轻的研究人员来培训研究生并组织会议，以便在优化和真实的代数几何社区之间进行富有成效的互动。该项目的第一部分侧重于从中心路径的角度研究SDO的复杂性的定量和算法问题。由于IPM在中心路径的邻域中运行，他们的效率受到中心路径的分析和代数几何性质的影响。研究人员探索了几种复杂性措施的基础上的程度，最坏情况下的收敛速度，和几何曲率的中心路径的正则和近不适定的情况。通过这些复杂性度量，特别是，一个可以定量证明复杂的IPM的特殊结构的严格互补条件的失败的情况下。项目的第二部分通过误差界和可行集的拓扑来研究PO的复杂性。与IPM不同，矩/平方和方法只处理一系列目标值（而不是解决方案），这可能无法充分反映向最优集合的进展。其结果是，目前的复杂性界限从矩/平方和的方法是纯粹的algebraicwith不依赖于拓扑/几何的可行集。研究人员将探索基于拓扑的复杂性措施，允许包括Betti数的可行集，从而提高PO求解器更精确的复杂性估计。这将严格解释为什么迭代算法更有可能停止在一个PO问题的局部最优解，因为连接组件或可行集中的孔的数量增加。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。