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Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.
Convex functions --- Banach spaces --- Geometry, Non-Euclidean --- Convex functions. --- Banach spaces. --- Geometry, Non-Euclidean. --- Non-Euclidean geometry --- Geometry --- Parallels (Geometry) --- Functions of complex variables --- Generalized spaces --- Topology --- Functions, Convex --- Functions of real variables --- Foundations
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Over the past two decades there have been significant advances in the field of optimization. In particular, convex optimization has emerged as a powerful signal processing tool, and the variety of applications continues to grow rapidly. This book, written by a team of leading experts, sets out the theoretical underpinnings of the subject and provides tutorials on a wide range of convex optimization applications. Emphasis throughout is on cutting-edge research and on formulating problems in convex form, making this an ideal textbook for advanced graduate courses and a useful self-study guide. Topics covered range from automatic code generation, graphical models, and gradient-based algorithms for signal recovery, to semidefinite programming (SDP) relaxation and radar waveform design via SDP. It also includes blind source separation for image processing, robust broadband beamforming, distributed multi-agent optimization for networked systems, cognitive radio systems via game theory, and the variational inequality approach for Nash equilibrium solutions.
Signal processing --- Mathematical optimization --- Convex functions --- Signal processing. --- Mathematical optimization. --- Convex functions. --- Functions, Convex --- Functions of real variables --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication)
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This book presents new achievements and results in the theory of conjugate duality for convex optimization problems. The perturbation approach for attaching a dual problem to a primal one makes the object of a preliminary chapter, where also an overview of the classical generalized interior point regularity conditions is given. A central role in the book is played by the formulation of generalized Moreau-Rockafellar formulae and closedness-type conditions, the latter constituting a new class of regularity conditions, in many situations with a wider applicability than the generalized interior point ones. The reader also receives deep insights into biconjugate calculus for convex functions, the relations between different existing strong duality notions, but also into several unconventional Fenchel duality topics. The final part of the book is consecrated to the applications of the convex duality theory in the field of monotone operators.
Convex functions. --- Duality theory (Mathematics). --- Mathematical optimization. --- Convex functions --- Duality theory (Mathematics) --- Mathematical optimization --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Monotone operators. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Functions, Convex --- Mathematics. --- Operations research. --- Decision making. --- Mathematical analysis. --- Analysis (Mathematics). --- System theory. --- Management science. --- Operations Research, Management Science. --- Operation Research/Decision Theory. --- Optimization. --- Systems Theory, Control. --- Analysis. --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Operator theory --- Algebra --- Topology --- Functions of real variables --- Systems theory. --- Global analysis (Mathematics). --- Operations Research/Decision Theory. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- 517.1 Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving --- Quantitative business analysis --- Statistical decision --- Philosophy --- Decision making
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The symposium focused on novel applications of methods from the calculus of variations to the solid mechanics of materials, including the development of new material models as well as the advancement of the corresponding computational techniques. Specific emphasis was put on the treatment of materials posessing an inherent microstructure and thus exhibiting a behavior which fundamentally involves multiple scales. Among the topic treated in this volume are: Energy-based modeling of material microstructures; modeling of the evolution of material microstructures; micromechanical modeling of shape-memory alloys; variational multiscale methods and associated numerical procedures; and micromechanics of multifield and multiphysics problems.
Materials -- Mechanical properties -- Congresses. --- Materials --- Engineering & Applied Sciences --- Chemical & Materials Engineering --- Applied Mathematics --- Materials Science --- Mechanical properties --- Calculus of variations. --- Control theory. --- Mathematical optimization. --- Convex functions. --- Functions, Convex --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Isoperimetrical problems --- Variations, Calculus of --- Engineering. --- Mathematical analysis. --- Analysis (Mathematics). --- Mathematical models. --- Applied mathematics. --- Engineering mathematics. --- Continuum mechanics. --- Continuum Mechanics and Mechanics of Materials. --- Calculus of Variations and Optimal Control; Optimization. --- Mathematical Modeling and Industrial Mathematics. --- Analysis. --- Appl.Mathematics/Computational Methods of Engineering. --- Maxima and minima --- Mechanics of continua --- Elasticity --- Mechanics, Analytic --- Field theory (Physics) --- Engineering --- Engineering analysis --- Mathematical analysis --- Models, Mathematical --- Simulation methods --- 517.1 Mathematical analysis --- Construction --- Industrial arts --- Technology --- Mathematics --- Functions of real variables --- Operations research --- System analysis --- Dynamics --- Machine theory --- Mechanics. --- Mechanics, Applied. --- Global analysis (Mathematics). --- Solid Mechanics. --- Mathematical and Computational Engineering. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Quantum theory --- Solids. --- Calculus of Variations and Optimization. --- Mathematical and Computational Engineering Applications. --- Data processing. --- Solid state physics --- Transparent solids
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