Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

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Linear Matrix Inequalities in System and Control Theory

The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

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Graphical Models, Exponential Families, and Variational Inference

Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field. He gives clear, lucid explanations of key results and ideas, with intuitive proofs, and provides critical examples and numerous illustrations to help elucidate the algorithms. Many of the results presented have been simplified and new insights provided. Of interest to theoretical computer scientists, operations researchers, and discrete mathematicians.

Approximation Algorithms

Monographs on statistics and applied probability

In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.

Semidefinite programming

This book develops geometric techniques for proving the polynomial time solvability of problems in convexity theory, geometry, and - in particular - combinatorial optimization. It offers a unifying approach based on two fundamental geometric algorithms: - the ellipsoid method for finding a point in a convex set and - the basis reduction method for point lattices. The ellipsoid method was used by Khachiyan to show the polynomial time solvability of linear programming. The basis reduction method yields a polynomial time procedure for certain diophantine approximation problems. A combination of these techniques makes it possible to show the polynomial time solvability of many questions concerning poyhedra - for instance, of linear programming problems having possibly exponentially many inequalities. Utilizing results from polyhedral combinatorics, it provides short proofs of the poynomial time solvability of many combinatiorial optimization problems. For a number of these problems, the geometric algorithms discussed in this book are the only techniques known to derive polynomial time solvability. This book is a continuation and extension of previous research of the authors for which they received the Fulkerson Prize, awarded by the Mathematical Programming Society and the American Mathematical Society.

Geometric Algorithms and Combinatorial Optimization

Part 1 Structures: graphs - basic graph theory - paths and circuits, J.A. Bondy, connectivity and network flows, A. Frank, matchings and extensions, W.R. Pulleyblank, colouring, stable sets and perfect graphs, B. Toft, embeddings and minors, C. Thomassen, random graphs, M. Karonski finite sets and relations - hypergraphs, P. Duchet, partially ordered sets, W.T. Trotter matroids - matroids - fundamental concepts, D.J.A. Welsh, matroid minors, P.D. Seymour, matroid optimization and algorithms, R.E. Bixby and W.H. Cunningham symmetric structures - permutation groups, P.J. Cameron, finite geometries, P.J. Cameron, block designs, A.E. Brouwer, association schemes, A.E. Brouwer and W. Haemers, codes, J.H. van Lint combinatorial structures in geometry and number theory - extremal problems in combinatorial geometry, P. Erdos and G. Purdy, convex polytopes and related complexes, V. Klee and P. Kleinschmidt, point lattices, J.C. Lagarias, combinatorial number theory, C. Pomerance and A. Sarkozy. Part 2 Aspects: algebraic enumeration, I.M. Gessel and R.P. Stanley asymptotic enumeration methods, A.M. Odlyzko extremal graph theory, B. Bollobas extremal set systems, P. Frankl Ramsey theory, J. Nesetril discrepancy theory, J. Beck and V.T. Sos automorphism groups, isomorphism, reconstruction, L. Babai optimization, M. Grotschel and L. Lovasz computational complexity, D.B. Shmoys and E. Tardos. Part 3 Methods: polyhedral combinatorics, A. Schrijver tools from linear algebra, C.D. Godsil tools from higher algebra, N. Alon probabilistic methods, J. Spencer topological methods, A. Bjorner. Part 4 Applications: combinatorics in operations research, A. Kolen and J.K. Lenstra combinatorics in electrical engineering and statics, A. Recski combinatorics in statistical mechanics, C.D. Godsil et al combinatorics in chemistry, D.H. Rouvray applications of combinatorics to molecular biology, M.S. Waterman combinatorics in computer science, L. Lovasz et al combinatorics in pure mathematics, L. Lovasz et al. Part 5 Horizons: infinite combinatorics, A. Hajnal combinatorial games, R.K. Guy the history of combinatorics, N.L. Biggs et al.

Handbook of Combinatorics

We study the problem of finding ground states of spin glasses with exterior magnetic field, and the problem of minimizing the number of vias holes on a printed circuit board, or contacts on a chip subject to pin preassignments and layer preferences. The former problem comes up in solid-state physics, and the latter in very-large-scale-integrated VLSI circuit design and in printed circuit board design. Both problems can be reduced to the max-cut problem in graphs. Based on a partial characterization of the cut polytope, we design a cutting plane algorithm and report on computational experience with it. Our method has been used to solve max-cut problems on graphs with up to 1,600 nodes.

An application of combinatorial optimization to statistical physics and circuit layout design

We show that the weighted versions of the stable set problem, the clique problem, the coloring problem and the clique covering problem are solvable in polynomial time for perfect graphs. Our algorithms are based on the ellipsoid method and a polynomial time separation algorithm for a certain class of positive semidefinite matrices related to Lovasz's bound θ(G) on the Shannon capacity of a graph. We show that θG) can be computed in polynomial time for all graphs G and also give a new characterization of perfect graphs in terms of this number θ(G). In addition we prove that the problem of verifying that a graph is imperfect is in NP. Moreover, we show that the computation of the stability number and the fractional stability number of a graph are unrelated with respect to hardness (if P ≠NP).

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Polynomial Algorithms for Perfect Graphs

In this paper we consider a clustering problem that arises in qualitative data analysis. This problem can be transformed to a combinatorial optimization problem, the clique partitioning problem. We have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope. These theoretical results are utilized in this paper. We describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before. We discuss some details of the implementation of our code and present our computational results. We mention applications from, e.g., zoology, economics, and the political sciences.

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Martin Grötschel

Papers

Geometric Algorithms and Combinatorial Optimization

Handbook of Combinatorics

An application of combinatorial optimization to statistical physics and circuit layout design

Polynomial Algorithms for Perfect Graphs

A cutting plane algorithm for a clustering problem