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This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Matrices. --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Matrices --- 512.64 --- 512.64 Linear and multilinear algebra. Matrix theory --- Linear and multilinear algebra. Matrix theory --- Addition. --- Analytic continuation. --- Arithmetic mean. --- Banach space. --- Binomial theorem. --- Block matrix. --- Bochner's theorem. --- Calculation. --- Cauchy matrix. --- Cauchy–Schwarz inequality. --- Characteristic polynomial. --- Coefficient. --- Commutative property. --- Compact space. --- Completely positive map. --- Complex number. --- Computation. --- Continuous function. --- Convex combination. --- Convex function. --- Convex set. --- Corollary. --- Density matrix. --- Diagonal matrix. --- Differential geometry. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence relation. --- Existential quantification. --- Extreme point. --- Fourier transform. --- Functional analysis. --- Fundamental theorem. --- G. H. Hardy. --- Gamma function. --- Geometric mean. --- Geometry. --- Hadamard product (matrices). --- Hahn–Banach theorem. --- Harmonic analysis. --- Hermitian matrix. --- Hilbert space. --- Hyperbolic function. --- Infimum and supremum. --- Infinite divisibility (probability). --- Invertible matrix. --- Lecture. --- Linear algebra. --- Linear map. --- Logarithm. --- Logarithmic mean. --- Mathematics. --- Matrix (mathematics). --- Matrix analysis. --- Matrix unit. --- Metric space. --- Monotonic function. --- Natural number. --- Open set. --- Operator algebra. --- Operator system. --- Orthonormal basis. --- Partial trace. --- Positive definiteness. --- Positive element. --- Positive map. --- Positive semidefinite. --- Positive-definite function. --- Positive-definite matrix. --- Probability measure. --- Probability. --- Projection (linear algebra). --- Quantity. --- Quantum computing. --- Quantum information. --- Quantum statistical mechanics. --- Real number. --- Riccati equation. --- Riemannian geometry. --- Riemannian manifold. --- Riesz representation theorem. --- Right half-plane. --- Schur complement. --- Schur's theorem. --- Scientific notation. --- Self-adjoint operator. --- Sign (mathematics). --- Special case. --- Spectral theorem. --- Square root. --- Standard basis. --- Summation. --- Tensor product. --- Theorem. --- Toeplitz matrix. --- Unit vector. --- Unitary matrix. --- Unitary operator. --- Upper half-plane. --- Variable (mathematics).

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Research on interior-point methods (IPMs) has dominated the field of mathematical programming for the last two decades. Two contrasting approaches in the analysis and implementation of IPMs are the so-called small-update and large-update methods, although, until now, there has been a notorious gap between the theory and practical performance of these two strategies. This book comes close to bridging that gap, presenting a new framework for the theory of primal-dual IPMs based on the notion of the self-regularity of a function. The authors deal with linear optimization, nonlinear complementarity problems, semidefinite optimization, and second-order conic optimization problems. The framework also covers large classes of linear complementarity problems and convex optimization. The algorithm considered can be interpreted as a path-following method or a potential reduction method. Starting from a primal-dual strictly feasible point, the algorithm chooses a search direction defined by some Newton-type system derived from the self-regular proximity. The iterate is then updated, with the iterates staying in a certain neighborhood of the central path until an approximate solution to the problem is found. By extensively exploring some intriguing properties of self-regular functions, the authors establish that the complexity of large-update IPMs can come arbitrarily close to the best known iteration bounds of IPMs. Researchers and postgraduate students in all areas of linear and nonlinear optimization will find this book an important and invaluable aid to their work.

Interior-point methods. --- Mathematical optimization. --- Programming (Mathematics). --- Mathematical optimization --- Interior-point methods --- Programming (Mathematics) --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Mathematical programming --- Goal programming --- Algorithms --- Functional equations --- Operations research --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Simulation methods --- System analysis --- 519.85 --- 681.3*G16 --- 681.3*G16 Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- 519.85 Mathematical programming --- Accuracy and precision. --- Algorithm. --- Analysis of algorithms. --- Analytic function. --- Associative property. --- Barrier function. --- Binary number. --- Block matrix. --- Combination. --- Combinatorial optimization. --- Combinatorics. --- Complexity. --- Conic optimization. --- Continuous optimization. --- Control theory. --- Convex optimization. --- Delft University of Technology. --- Derivative. --- Differentiable function. --- Directional derivative. --- Division by zero. --- Dual space. --- Duality (mathematics). --- Duality gap. --- Eigenvalues and eigenvectors. --- Embedding. --- Equation. --- Estimation. --- Existential quantification. --- Explanation. --- Feasible region. --- Filter design. --- Function (mathematics). --- Implementation. --- Instance (computer science). --- Invertible matrix. --- Iteration. --- Jacobian matrix and determinant. --- Jordan algebra. --- Karmarkar's algorithm. --- Karush–Kuhn–Tucker conditions. --- Line search. --- Linear complementarity problem. --- Linear function. --- Linear programming. --- Lipschitz continuity. --- Local convergence. --- Loss function. --- Mathematician. --- Mathematics. --- Matrix function. --- McMaster University. --- Monograph. --- Multiplication operator. --- Newton's method. --- Nonlinear programming. --- Nonlinear system. --- Notation. --- Operations research. --- Optimal control. --- Optimization problem. --- Parameter (computer programming). --- Parameter. --- Pattern recognition. --- Polyhedron. --- Polynomial. --- Positive semidefinite. --- Positive-definite matrix. --- Quadratic function. --- Requirement. --- Result. --- Scientific notation. --- Second derivative. --- Self-concordant function. --- Sensitivity analysis. --- Sign (mathematics). --- Signal processing. --- Simplex algorithm. --- Simultaneous equations. --- Singular value. --- Smoothness. --- Solution set. --- Solver. --- Special case. --- Subset. --- Suggestion. --- Technical report. --- Theorem. --- Theory. --- Time complexity. --- Two-dimensional space. --- Upper and lower bounds. --- Variable (computer science). --- Variable (mathematics). --- Variational inequality. --- Variational principle. --- Without loss of generality. --- Worst-case complexity. --- Yurii Nesterov. --- Mathematical Optimization --- Mathematics --- Programming (mathematics)