Semidefinite programming algorithms for convex optimization over nonnegative polynomials with applications in control and signal processing.

非负多项式上凸优化的半定规划算法及其在控制和信号处理中的应用在过去的15年中，半定规划已发展成为各种工程应用，特别是在控制和信号处理中的一种重要的数值工具。与此同时，semide.nite编程算法的进步已经产生了几个高质量和免费的semide.nite编程软件包。 这些通用求解器利用一些（稀疏）问题结构，并可以解决相当大的问题实例。然而，对于控制和信号处理应用，这往往涉及矩阵变量和密集的线性矩阵不等式约束，仍然存在着最佳的求解器的能力和实际应用的要求之间的巨大差距。本建议的主要目标是缩小这一差距，通过开发快速算法的一些speci.c类的半成品。nite程序是重要的控制和信号处理。 该方案的重点是半夜场程序，其中系数矩阵可以表示为一小部分低秩矩阵的不同线性组合。这种类型的结构是常见的，并且包括从非负多项式的平方和表示导出的一类重要的半成品程序。该项目的第二个目标是研究半现场编程在图形建模和时间序列分析中的新应用，特别是对变量对具有条件独立约束的线性模型的估计。（非负多项式）、数值分析（离散变换和正交多项式）和最优化（边界点法）。更广泛的影响将免费提供基于研究成果的软件，这将有助于更广泛地采用半成品程序设计技术。 研究结果将被整合到加州大学洛杉矶分校电气工程系的研究生优化序列中。我们还计划通过暑期实习的方式提供本科生的研究机会。本科生研究人员将通过加州大学洛杉矶分校的工程与多样性卓越中心（CEED）招募，目标是增加对研究生学习感兴趣的CEED本科生的数量。

Semidefinite programming algorithms for convex optimization over nonnegative polynomials with applications in control andsignal processing During the last .fteen years semide.nite programming has developed into an important numerical tool for a variety of engineering applications, in particular in control and signal processing. At the same time, advances in semide.nite programming algorithms have resulted in several high-quality and freely available semide.nite programming software packages. These general-purpose solvers exploit some (sparse) problem structure, and can solve fairly large problem instances. However, for control and signal processing applications, which often involve matrix variables and dense linear matrix inequality constraints, there remains a substantial gap between the capabilities of the best solvers and the requirements for applications in practice. The main goal of this proposal is to narrow this gap, by developing fast algorithms for some speci.c classes of semide.nite programs that are important in control and signal processing. The proposal focuses on semide.nite programs in which the coe.cient matrices can be expressed as di.erent linear combinations of a small set of low-rank matrices. This type ofstructure is common and includes the important class of semide.nite programs derived from sum-of-squares representations of nonnegative polynomials. A second goal of the project is to investigate new applications of semide.nite programming to graphical modeling and time series analysis, and in particular the estimation of linear models with conditional independence constraints on pairs of variables.Intellectual merit The project will result in improved algorithms and software for solving important classes of semide.nite programming problems, by combining techniques from system theory (nonnegative polynomials), numerical analysis (discrete transforms and orthogonal polynomials) and optimization (interior-point methods).Broader impacts Software based on the research results will be made freely available, which will contribute to a more widespread adoption of semide.nite programming techniques. The results will be integrated in the graduate optimization sequence in the Electrical Engineering Department at UCLA. We also plan to o.er undergraduate research opportunitiesvia summer internships. The undergraduate student researchers will be recruited through the Center of Excellence in Engineering and Diversity (CEED) at UCLA, with the goal ofincreasing the number of CEED undergraduates interested in graduate study.