Download or read book Elementary Convexity with Optimization written by Vivek S. Borkar and published by Springer Nature. This book was released on 2023-06-26 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines.
Download or read book Variational Calculus with Elementary Convexity written by J.L. Troutman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: The calculus of variations, whose origins can be traced to the works of Aristotle and Zenodoros, is now Ii vast repository supplying fundamental tools of exploration not only to the mathematician, but-as evidenced by current literature-also to those in most branches of science in which mathematics is applied. (Indeed, the macroscopic statements afforded by variational principles may provide the only valid mathematical formulation of many physical laws. ) As such, it retains the spirit of natural philosophy common to most mathematical investigations prior to this century. How ever, it is a discipline in which a single symbol (b) has at times been assigned almost mystical powers of operation and discernment, not readily subsumed into the formal structures of modern mathematics. And it is a field for which it is generally supposed that most questions motivating interest in the subject will probably not be answerable at the introductory level of their formulation. In earlier articles,1,2 it was shown through several examples that a complete characterization of the solution of optimization problems may be available by elementary methods, and it is the purpose of this work to explore further the convexity which underlay these individual successes in the context of a full introductory treatment of the theory of the variational calculus. The required convexity is that determined through Gateaux variations, which can be defined in any real linear space and which provide an unambiguous foundation for the theory.
Download or read book Variational Calculus and Optimal Control written by John L. Troutman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the variational methods used to formulate and solve mathematical and physical problems, allowing the reader an insight into the systematic use of elementary (partial) convexity of differentiable functions in Euclidian space. By helping students directly characterize the solutions for many minimization problems, the text serves as a prelude to the field theory for sufficiency, laying as it does the groundwork for further explorations in mathematics, physics, mechanical and electrical engineering, as well as computer science.
Download or read book Convex Optimization written by Stephen P. Boyd and published by Cambridge University Press. This book was released on 2004-03-08 with total page 744 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
Download or read book Convexity and Optimization in Finite Dimensions I written by Josef Stoer and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dantzig's development of linear programming into one of the most applicable optimization techniques has spread interest in the algebra of linear inequalities, the geometry of polyhedra, the topology of convex sets, and the analysis of convex functions. It is the goal of this volume to provide a synopsis of these topics, and thereby the theoretical back ground for the arithmetic of convex optimization to be treated in a sub sequent volume. The exposition of each chapter is essentially independent, and attempts to reflect a specific style of mathematical reasoning. The emphasis lies on linear and convex duality theory, as initiated by Gale, Kuhn and Tucker, Fenchel, and v. Neumann, because it represents the theoretical development whose impact on modern optimi zation techniques has been the most pronounced. Chapters 5 and 6 are devoted to two characteristic aspects of duality theory: conjugate functions or polarity on the one hand, and saddle points on the other. The Farkas lemma on linear inequalities and its generalizations, Motzkin's description of polyhedra, Minkowski's supporting plane theorem are indispensable elementary tools which are contained in chapters 1, 2 and 3, respectively. The treatment of extremal properties of polyhedra as well as of general convex sets is based on the far reaching work of Klee. Chapter 2 terminates with a description of Gale diagrams, a recently developed successful technique for exploring polyhedral structures.
Download or read book Generalized Convexity and Optimization written by Alberto Cambini and published by Springer Science & Business Media. This book was released on 2008-10-14 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors have written a rigorous yet elementary and self-contained book to present, in a unified framework, generalized convex functions. The book also includes numerous exercises and two appendices which list the findings consulted.
Download or read book Variational Calculus with Elementary Convexity written by John L. Troutman and published by . This book was released on 1996 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Generalized Convexity and Optimization written by Alberto Cambini and published by Springer. This book was released on 2008-10-15 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors have written a rigorous yet elementary and self-contained book to present, in a unified framework, generalized convex functions. The book also includes numerous exercises and two appendices which list the findings consulted.
Download or read book Convex Optimization Algorithms written by Dimitri Bertsekas and published by Athena Scientific. This book was released on 2015-02-01 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive and accessible presentation of algorithms for solving convex optimization problems. It relies on rigorous mathematical analysis, but also aims at an intuitive exposition that makes use of visualization where possible. This is facilitated by the extensive use of analytical and algorithmic concepts of duality, which by nature lend themselves to geometrical interpretation. The book places particular emphasis on modern developments, and their widespread applications in fields such as large-scale resource allocation problems, signal processing, and machine learning. The book is aimed at students, researchers, and practitioners, roughly at the first year graduate level. It is similar in style to the author's 2009"Convex Optimization Theory" book, but can be read independently. The latter book focuses on convexity theory and optimization duality, while the present book focuses on algorithmic issues. The two books share notation, and together cover the entire finite-dimensional convex optimization methodology. To facilitate readability, the statements of definitions and results of the "theory book" are reproduced without proofs in Appendix B.
Download or read book Convex Analysis written by Ralph Tyrell Rockafellar and published by Princeton University Press. This book was released on 2015-04-29 with total page 470 pages. Available in PDF, EPUB and Kindle. Book excerpt: Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.
Download or read book Convex Functions and Optimization Methods on Riemannian Manifolds written by C. Udriste and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately "nowhere" to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real valued function whose restriction to every geodesic arc is convex.
Download or read book Convex Analysis for Optimization written by Jan Brinkhuis and published by Springer Nature. This book was released on 2020-05-05 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook offers graduate students a concise introduction to the classic notions of convex optimization. Written in a highly accessible style and including numerous examples and illustrations, it presents everything readers need to know about convexity and convex optimization. The book introduces a systematic three-step method for doing everything, which can be summarized as "conify, work, deconify". It starts with the concept of convex sets, their primal description, constructions, topological properties and dual description, and then moves on to convex functions and the fundamental principles of convex optimization and their use in the complete analysis of convex optimization problems by means of a systematic four-step method. Lastly, it includes chapters on alternative formulations of optimality conditions and on illustrations of their use. "The author deals with the delicate subjects in a precise yet light-minded spirit... For experts in the field, this book not only offers a unifying view, but also opens a door to new discoveries in convexity and optimization...perfectly suited for classroom teaching." Shuzhong Zhang, Professor of Industrial and Systems Engineering, University of Minnesota
Download or read book Abstract Convexity and Global Optimization written by Alexander M. Rubinov and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func tion. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.
Download or read book Undergraduate Convexity written by Niels Lauritzen and published by World Scientific. This book was released on 2013 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples.Starting from linear inequalities and FourierOCoMotzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the KarushOCoKuhnOCoTucker conditions, duality and an interior point algorithm.
Download or read book Convexity and Optimization in Banach Spaces written by Viorel Barbu and published by Springer Science & Business Media. This book was released on 2012-01-03 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: An updated and revised edition of the 1986 title Convexity and Optimization in Banach Spaces, this book provides a self-contained presentation of basic results of the theory of convex sets and functions in infinite-dimensional spaces. The main emphasis is on applications to convex optimization and convex optimal control problems in Banach spaces. A distinctive feature is a strong emphasis on the connection between theory and application. This edition has been updated to include new results pertaining to advanced concepts of subdifferential for convex functions and new duality results in convex programming. The last chapter, concerned with convex control problems, has been rewritten and completed with new research concerning boundary control systems, the dynamic programming equations in optimal control theory and periodic optimal control problems. Finally, the structure of the book has been modified to highlight the most recent progression in the field including fundamental results on the theory of infinite-dimensional convex analysis and includes helpful bibliographical notes at the end of each chapter.
Download or read book Lectures on Convex Geometry written by Daniel Hug and published by Springer Nature. This book was released on 2020-08-27 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.
Download or read book Convex Optimization Theory written by Dimitri Bertsekas and published by Athena Scientific. This book was released on 2009-06-01 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: An insightful, concise, and rigorous treatment of the basic theory of convex sets and functions in finite dimensions, and the analytical/geometrical foundations of convex optimization and duality theory. Convexity theory is first developed in a simple accessible manner, using easily visualized proofs. Then the focus shifts to a transparent geometrical line of analysis to develop the fundamental duality between descriptions of convex functions in terms of points, and in terms of hyperplanes. Finally, convexity theory and abstract duality are applied to problems of constrained optimization, Fenchel and conic duality, and game theory to develop the sharpest possible duality results within a highly visual geometric framework. This on-line version of the book, includes an extensive set of theoretical problems with detailed high-quality solutions, which significantly extend the range and value of the book. The book may be used as a text for a theoretical convex optimization course; the author has taught several variants of such a course at MIT and elsewhere over the last ten years. It may also be used as a supplementary source for nonlinear programming classes, and as a theoretical foundation for classes focused on convex optimization models (rather than theory). It is an excellent supplement to several of our books: Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2017), Network Optimization(Athena Scientific, 1998), Introduction to Linear Optimization (Athena Scientific, 1997), and Network Flows and Monotropic Optimization (Athena Scientific, 1998).
