Methods for Matrix Optimization Problems in Control and Statistical Signal Processing

ECS-9707111 Boyd 在过去的几年里，各种凸矩阵优化问题的强大的新方法已经出现。 这些研究大多集中在半定规划问题上，即半定规划问题。例如，在线性矩阵不等式（LMI）上最小化线性成本函数的问题。 SDPs是凸优化问题，可以使用最近开发的邻域点算法以很高的效率来解决。 这些矩阵优化方法在多个领域，特别是控制系统和组合优化中产生了直接的实际和理论影响。 在控制理论中，基本思想是将分析或综合问题用凸矩阵优化问题来表示，然后用数值方法求解。 关于LMI在控制中的研究现状可以总结如下： 有深入的研究，以确定控制问题，可以投在LMIS方面，和那些LM 1配方是不可能存在的。 在后一种情况下，双线性矩阵不等式（BMIS）已被认为是一个统一的形式。 数学规划和控制理论的结合使得求解SDPS的边界点算法取得了非常迅速的进展，重点是局部收敛速度，最坏情况的复杂性等，并将线性规划（LP）中复杂而有效的原-对偶邻域点法推广到SDP中。 SDP的边界点方法的几个基本软件实现已经可用。 事实证明，这些代码对于中小型问题很有用，但对于较大的问题来说往往太慢，因为它们很少或根本不利用问题结构。 LML正在成为控制的基本工具，就像Riccati方程在20世纪60年代成为基本工具一样。 因此，LMI/SDP求解器正在成为控制（计算和实践）基础设施的一部分，就像Riccati方程求解器现在一样。 同样的技术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。 在这里，在问题制定领域仍然存在许多挑战（即，识别可以在该框架中投射的统计信号处理问题），但是已经清楚的是，该方法将是有益的。 也许甚至比在控制中，矩阵优化在统计学中的实际应用将需要大规模问题的有效算法。 该提案汇集了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LMIS公式表示的问题，以及更强大算法的发展（和实用代码）的矩阵优化问题，出现在控制和统计信号处理。 研究工作将与强大的教育组成部分相结合，通过将这项工作与新开发的课程凸优化与工程应用相结合。 我们将统计信号处理的新研究成果直接转化为课程材料，这将有助于拓宽课程中提出的应用范围。 对数值方法的新研究将以两种重要的方式进入课程：首先，作为实施的高级主题（当前课程中缺少），其次，通过提供学生可以在课程中使用的工具，特别是在项目中。 另一个重要（但不太直接）的教育目标是以完全跨学科的方式培养两名博士生，使他们具备从鲁棒控制理论或统计到优化理论和大规模数值算法实现的技能。 随着计算能力的持续指数增长，具有这种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.