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

半定优化和多项式优化（SDO和PO）是具有重要理论和实践意义的课题，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。高效内点方法（ipm）的稳步发展使SDO作为一种新兴的计算工具在PO和量子计算中发挥了重要作用。SDO和PO的复杂性在比特计算模型中是众所周知的：目前还没有多项式时间算法来找到这类优化问题的精确最优解。然而，即使对于近似解，也存在ipm或PO的松弛层次不能解决的病态实例。鉴于这一挑战，显然需要通过更广泛的复杂性度量来研究复杂性。这种新颖的方法允许对高复杂性的情况进行更精细的分类。本项目通过真正的代数几何来解决SDO和PO复杂性方面的几个关键问题，从而实现上述目标。该项目的结果将增强对SDO和pod复杂性的理解，并有可能影响其他学科，包括量子信息科学，其中量子ipm的新兴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研究人员来培训研究生并组织会议，以便在优化和实际代数几何社区之间进行富有成效的互动。项目的第一部分从中心路径的角度关注关于SDO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，其效率受到中心路径的解析和代数几何性质的影响。研究人员探索了几种基于程度、最坏情况收敛率和中心路径的几何曲率的复杂性措施，用于正则和接近病态的实例。特别是，通过这些复杂性度量，可以定量地证明ipm在特殊结构表现出严格互补性条件失效的实例中的复杂性。第二部分通过误差界和可行集拓扑研究了PO的复杂度。与ipm不同，矩/平方和方法只处理一系列目标值（而不是解），这可能不能充分反映向最优集合的进展。因此，当前矩/平方和方法的复杂度界是纯代数的，不依赖于可行集的拓扑/几何。研究人员将探索基于拓扑的复杂性措施，允许包含可行集的贝蒂数，从而提高PO求解器更精确的复杂性估计。这将严格地解释为什么迭代算法更有可能在POproblem的局部最优处停止，因为可行集中连接组件的数量或孔的数量增加。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.