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Optimization and Dynamical Systems Optimization and Dynamical Systems Uwe Helmke1 John B. Moore2 2nd Edition March 1996 1. Department of Mathematics, University of W¨urzburg, D-97074 W¨urzburg, Germany. 2. Department of Systems Engineering and Cooperative Research Centre for Robust and Adaptive Systems, Research School of Information Sci- ences and Engineering, Australian National University, Canberra, ACT 0200, Australia. Preface This work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control sys- tems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emer- gence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems the- ory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical sys- tems implementation, either in continuous time or discrete time , which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geomet- ric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods. Even for such problems that do admit solutions via algebraic methods, as for example the classical task of singular value decomposition, there is merit in viewing the task as a certain matrix optimization problem, so as to shift the focus from algebraic methods to geometric methods. It is in this context that gradient flows on manifolds appear as a natural approach to achieve construction methods that complement the existence and uniqueness results of geometric invari- iv Preface ant theory. There has been an attempt to bridge the disciplines of engineering and mathematics in such a way that the work is lucid for engineers and yet suitably rigorous for mathematicians. There is also an attempt to teach, to provide insight, and to make connections, rather than to present the results as a fait accompli. The research for this work has been carried out by the authors while visiting each other’s institutions. Some of the work has been written in conjunction with the writing of research papers in collaboration with PhD students Jane Perkins and Robert Mahony, and post doctoral fellow Weiy- ong Yan. Indeed, the papers benefited from the book and vice versa, and consequently many of the paragraphs are common to both. Uwe Helmke has a background in global analysis with a strong interest in systems theory, and John Moore has a background in control systems and signal processing. Acknowledgements This work was partially supported by grants from Boeing Commercial Air- plane Company, the Cooperative Research Centre for Robust and Adap- tive Systems, and the German-Israeli Foundation. The LATEXsetting of this manuscript has been expertly achieved by Ms Keiko Numata. Assistance came also from Mrs Janine Carney, Ms Lisa Crisafulli, Mrs Heli Jack- son, Mrs Rosi Bonn, and Ms Marita Rendina. James Ashton, Iain Collings, and Jeremy Matson lent LATEXprogramming support. The simulations have been carried out by our PhD students Jane Perkins and Robert Mahony, and post-doctoral fellow Weiyong Yan. Diagrams have been prepared by Anton Madievski. Bob Mahony was also a great help in proof reading and polishing the manuscript. Finally, it is a pleasure to thank Anthony Bloch, Roger Brockett and Leonid Faybusovich for helpful discussions. Their ideas and insights had a crucial influence on our way of thinking. Foreword By Roger W. Brockett Differential equations have provided the language for expressing many of the important ideas of automatic control. Questions involving optimiza- tion, dynamic compensation and stability are effectively expressed in these terms. However, as technological advances in VLSI reduce the cost of com- puting, we are seeing a greater emphasis on control problems that involve real-time computing. In some cases this means using a digital implemen- tation of a conventional linear controller, but in other cases the additional flexibility that digital implementation allows is being used to create sys- tems that are of a completely different character. These developments have resulted in a need for effective ways to think about systems which use both ordinary dynamical compensation and logical rules to generate feedback signals. Until recently the dynamic response of systems that incorporate if-then rules, branching, etc. has not been studied in a very effective way. From this latter point of view it is useful to know that there exist families of ordinary differential equations whose flows are such as to generate a sorting of the numerical values of the various components of the initial conditions, solve a linear programming problem, etc. A few years ago, I observed that a natural formulation of a steepest descent algorithm for solving a least-squares matching problem leads to a differential equation for carrying out such operations. During the course of some conversation’s with Anthony Bloch it emerged that there is a simplified version of the matching equations, obtained by recasting them as flows on the Lie algebra and then restricting them to a subspace, and that this simplified version can be used to sort lists as well. Bloch observed that the restricted equations are identical to the Toda lattice equations in the form introduced by Hermann vi Foreword Flaschka nearly twenty years ago. In fact, one can see in J¨urgen Moser’s early paper on the solution of the Toda Lattice, a discussion of sorting couched in the language of scattering theory and presumably one could have developed the subject from this, rather different, starting point. The fact that a sorting flow can be viewed as a gradient flow and that each simple Lie algebra defines a slightly different version of this gradient flow was the subject of a systematic analysis in a paper that involved collaboration with Bloch and Tudor Ratiu. The differential equations for matching referred to above are actually formulated on a compact matrix Lie group and then rewritten in terms of matrices that evolve in the Lie algebra associated with the group. It is only with respect to a particular metric on a smooth submanifold of this Lie algebra (actually the manifold of all matrices having a given set of eigen- values) that the final equations appear to be in gradient form. However, as equations on this manifold, the descent equations can be set up so as to find the eigenvalues of the initial condition matrix. This then makes contact with the subject of numerical linear algebra and a whole line of interesting work going back at least to Rutishauser in the mid 1950s and continuing to the present day. In this book Uwe Helmke and John Moore emphasize prob- lems, such as computation of eigenvalues, computation of singular values, construction of balanced realizations, etc. involving more structure than just sorting or solving linear programming problems. This focus gives them a natural vehicle to introduce and interpret the mathematical aspects of the subject. A recent Harvard thesis by Steven Smith contains a detailed discussion of the numerical performance of some algorithms evolving from this point of view. The circle of ideas discussed in this book have been developed in some other directions as well. Leonid Faybusovich has taken a double bracket equation as the starting point in a general approach to interior point meth- ods for linear programming. Wing Wong and I have applied this type of thinking to the assignment problem, attempting to find compact ways to formulate a gradient flow leading to the solution. Wong has also exam- ined similar methods for nonconvex problems and Jeffrey Kosowsky has compared these methods with other flow methods inspired by statistical physics. In a joint paper with Bloch we have investigated certain partial differential equation models for sorting continuous functions, i.e. generating the monotone equi-measurable rearrangements of functions, and Saveliev has examined a family of partial differential equations of the double bracket type, giving them a cosmological interpretation. It has been interesting to see how rapidly the literature in this area has grown. The present book comes at a good time, both because it provides a well reasoned introduction to the basic ideas for those who are curious Foreword vii and because it provides a self-contained account of the work on balanced realizations worked out over the last few years by the authors. Although I would not feel comfortable attempting to identify the most promising direction for future work, all indications are that this will continue to be a fruitful area for research. Contents Preface iii Foreword v 1 Matrix Eigenvalue Methods 1 1.1Introduction........................... 1 1.2 Power Method for Diagonalization . 5 1.3 The Rayleigh Quotient Gradient Flow . 14 1.4TheQRAlgorithm....................... 30 1.5 Singular Value Decomposition (SVD) . 35 1.6 Standard Least Squares Gradient Flows . 38 2 Double Bracket Isospectral Flows 43 2.1 Double Bracket Flows for Diagonalization . 43 2.2TodaFlowsandtheRiccatiEquation............ 58 2.3 Recursive Lie-Bracket Based Diagonalization . 68 3 Singular Value Decomposition 81 3.1SVDviaDoubleBracketFlows................ 81 3.2 A Gradient Flow Approach to SVD . 84 4 Linear Programming 101 4.1 The RˆoleofDoubleBracketFlows.............. 101 4.2InteriorPointFlowsonaPolytope.............. 111 4.3 Recursive Linear Programming/Sorting .
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