Methods for Matrix Optimization Problems in Control and Statistical Signal Processing

近几年来，求解各种凸矩阵优化问题的新方法层出不穷，如ECS-9707111，Boyd。这方面的研究大多集中在半定规划问题(SDP)上，即在线性矩阵不等式(LMI)上最小化线性代价函数的问题。SDP是凸优化问题，可以使用最近发展的内点算法以极高的效率求解。这些矩阵优化方法在几个领域产生了直接的实践和理论影响，特别是控制系统和组合优化。在控制理论中，基本思想是将分析或综合问题表示为凸矩阵优化问题，然后用数值方法求解。LMI在控制中的研究现状可以概括为：已经有大量的研究来识别可以用LMI表示的控制问题，以及那些不太可能存在LM1公式的控制问题。在后一种情况下，双线性矩阵不等式(BMI)被认为是统一的形式。数学规划和控制理论的结合使得求解SDPS的内点算法取得了非常迅速的进展，主要集中在局部收敛速度、最坏情况的复杂性等方面，并将为线性规划(LP)开发的复杂而有效的原始-对偶内点方法扩展到SDP。用于SDP的内点法的几个基本软件实现已经可用。事实证明，这些代码对中小型问题很有用，但对于较大的问题往往太慢，因为它们利用的问题结构很少或根本没有问题结构。LML正在成为控制的基本工具，就像里卡蒂方程在20世纪60年代成为基本工具一样。因此，LMI/SDP解算器正在成为控制(计算和实践)基础设施的一部分，就像Riccati方程解算器现在一样。同样的技术也可以应用于其他几个领域。例如，在统计信号处理中，SDP允许定义涉及协方差矩阵的表示和分解的优化问题。在这方面，在问题制定领域仍存在许多挑战(即确定可在这一框架中提出的统计信号处理问题)，但已经很明显，这种方法将是有益的。在统计学中，矩阵优化的实际应用可能比控制更需要高效的算法来解决大规模问题。这一建议汇集了涉及三个关键领域的研究人员：控制理论、统计学和大规模数值优化，以解决下一个逻辑研究领域：识别统计信号处理中可以用LMI表示的问题，并为控制和统计信号处理中出现的矩阵优化问题开发更强大的算法(和实用代码)。通过将研究成果与新开发的课程《凸优化与工程应用》相结合，研究工作将得到强有力的教育内容的补充。我们将把统计信号处理方面的新研究成果直接转化为课程材料，这将有助于拓宽本课程的应用范围。有关数值方法的新研究将以两种重要方式进入本课程：第一，作为实施的高级主题(当前课程中缺失)，第二，通过提供学生在课程中可以使用的工具，特别是在项目中。另一个重要(但不那么直接)的教育目标是以一种完全跨学科的方式培养两名博士生，让他们掌握从稳健控制理论或统计学到优化理论和大规模数值算法实施的各种技能。随着计算能力继续呈指数级增长，具有这一背景的研究人员将越来越有价值。

ECS-9707111 Boyd Powerful new methods for various convex matrix optimization problems have emerged over the last few years. Most of this research has concentrated on the semidefinite programming problem (SDP), i. e., the problem of minimizing a linear cost function over linear matrix inequalities (LMIs). SDPs are convex optimization problems that can be solved with great efficiency using recently developed interior-point algorithms. These matrix optimization methods have had immediate practical and theoretical impact in several fields, notably control systems and combinatorial optimization. In control theory, the basic idea is to formulate the analysis or synthesis problem in terms of convex matrix optimization problems, which are then solved numerically. The current state of research on LMIs in control can be summarized: There has been intensive research on identifying control problems that can be cast in terms of LMIS, and those for which an LM1 formulation is unlikely to exist. In the latter case, bilinear matrix inequalities (BMIS) have been recognized as a unifying form. The combined activity in mathematical programming and control theory has led to very rapid progress in interior-point algorithms for solving SDPS, focusing on local convergence rates, worst-case complexity, etc., and on extending to SDP the sophisticated and efficient primal-dual interior-point methods developed for linear programming (LP). Several basic software implementations of interior-point methods for SDP have become available. These codes have proven useful for small to medium-sized problems, but tend to be too slow for larger problems, since they exploit little or no problem structure. LMls are becoming basic tools in control, much the way Riccati equations became basic tools in the 1960s. Thus, LMI/SDP solvers are becoming part of the infrastructure of control (computation and practice), just as Riccati equation solvers are now. The same techniques can be applied in several other fields. In statistical signal processing, for example, SDP allows one to define optimization problems that involve representation and decomposition of covariance matrices. Here many challenges are still to be found in the problem formulation area (i.e., identifying statistical signal processing problems that can be cast in this framework) but it is already clear that the approach will be rewarding. Perhaps even more than in control, practical use of matrix optimization in statistics will require efficient algorithms for large-scale problems. This proposal brings together researchers in the three key areas involved: control theory, statistics, and large-scale numerical optimization, in order to address the next logical areas of research: identification of the problems in statistical signal processing that can be formulated in terms of LMIS, and the development of more powerful algorithms (and practical codes) for the matrix optimization problems that arise in control and statistical signal processing. The research effort will be complemented with a strong educational component, by integrating the effort with the newly developed course Convex Optimization with Engineering Applications. We will transition new research results in statistical signal processing directly into the course material, which will help broaden the range of applications presented in the course. The new research on numerical methods will enter the course in two important ways: first, as an advanced topic on implementation (missing from the current course), and second, by providing tools that students can use during the course, especially in projects. Another important (but less direct) educational goal is to train two PhD students in a completely interdisciplinary fashion, equipping them with skills that range from, say, robust control theory or statistics, to optimization theory and implementation of large-scale numerical algorithms. As the power of computing continues its exponential rise, researchers with this background will be increasingly valuable.