Topics In Nonconvex Optimization
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| Author | : Shashi K. Mishra |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 276 |
| Release | : 2011-05-21 |
| Genre | : Business & Economics |
| ISBN | : 1441996400 |
Nonconvex Optimization is a multi-disciplinary research field that deals with the characterization and computation of local/global minima/maxima of nonlinear, nonconvex, nonsmooth, discrete and continuous functions. Nonconvex optimization problems are frequently encountered in modeling real world systems for a very broad range of applications including engineering, mathematical economics, management science, financial engineering, and social science. This contributed volume consists of selected contributions from the Advanced Training Programme on Nonconvex Optimization and Its Applications held at Banaras Hindu University in March 2009. It aims to bring together new concepts, theoretical developments, and applications from these researchers. Both theoretical and applied articles are contained in this volume which adds to the state of the art research in this field. Topics in Nonconvex Optimization is suitable for advanced graduate students and researchers in this area.
| Author | : Prateek Jain |
| Publisher | : Foundations and Trends in Machine Learning |
| Total Pages | : 218 |
| Release | : 2017-12-04 |
| Genre | : Machine learning |
| ISBN | : 9781680833683 |
Non-convex Optimization for Machine Learning takes an in-depth look at the basics of non-convex optimization with applications to machine learning. It introduces the rich literature in this area, as well as equips the reader with the tools and techniques needed to apply and analyze simple but powerful procedures for non-convex problems. Non-convex Optimization for Machine Learning is as self-contained as possible while not losing focus of the main topic of non-convex optimization techniques. The monograph initiates the discussion with entire chapters devoted to presenting a tutorial-like treatment of basic concepts in convex analysis and optimization, as well as their non-convex counterparts. The monograph concludes with a look at four interesting applications in the areas of machine learning and signal processing, and exploring how the non-convex optimization techniques introduced earlier can be used to solve these problems. The monograph also contains, for each of the topics discussed, exercises and figures designed to engage the reader, as well as extensive bibliographic notes pointing towards classical works and recent advances. Non-convex Optimization for Machine Learning can be used for a semester-length course on the basics of non-convex optimization with applications to machine learning. On the other hand, it is also possible to cherry pick individual portions, such the chapter on sparse recovery, or the EM algorithm, for inclusion in a broader course. Several courses such as those in machine learning, optimization, and signal processing may benefit from the inclusion of such topics.
| Author | : Ying Cui |
| Publisher | : Society for Industrial and Applied Mathematics (SIAM) |
| Total Pages | : 0 |
| Release | : 2022 |
| Genre | : Convex functions |
| ISBN | : 9781611976731 |
"This monograph serves present and future needs where nonconvexity and nondifferentiability are inevitably present in the faithful modeling of real-world applications of optimization"--
| Author | : F.H. Clarke |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 481 |
| Release | : 2013-11-11 |
| Genre | : Science |
| ISBN | : 1475760191 |
This volume contains the edited texts of the lect. nres presented at the International School of Mathematics devoted to Nonsmonth Optimization, held from . June 20 to July I, 1988. The site for the meeting was the "Ettore ~Iajorana" Centre for Sci entific Culture in Erice, Sicily. In the tradition of these meetings the main purpose was to give the state-of-the-art of an important and growing field of mathematics, and to stimulate interactions between finite-dimensional and infinite-dimensional op timization. The School was attended by approximately 80 people from 23 countries; in particular it was possible to have some distinguished lecturers from the SO\·iet Union, whose research institutions are here gratt-fnlly acknowledged. Besides the lectures, several seminars were delivered; a special s·~ssion was devoted to numerical computing aspects. The result was a broad exposure. gi ·. ring a deep knowledge of the present research tendencies in the field. We wish to express our appreciation to all the participants. Special mention 5hould be made of the Ettorc ;. . Iajorana Centre in Erice, which helped provide a stimulating and rewarding experience, and of its staff which was fundamental for the success of the meeting. j\, loreover, WP want to extend uur deep appreci
| Author | : David Yang Gao |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 476 |
| Release | : 2000-01-31 |
| Genre | : Mathematics |
| ISBN | : 9780792361459 |
Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics. Topics covered include the classical (minimax) mono-duality of convex static equilibria, the beautiful bi-duality in dynamical systems, the interesting tri-duality in non-convex problems and the complicated multi-duality in general canonical systems. A potentially powerful sequential canonical dual transformation method for solving fully nonlinear problems is developed heuristically and illustrated by use of many interesting examples as well as extensive applications in a wide variety of nonlinear systems, including differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, non-smooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and non-convex dynamical systems. With exceptionally coherent and lucid exposition, the work fills a big gap between the mathematical and engineering sciences. It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science. Much of the book contains material that is new, both in its manner of presentation and in its research development. A self-contained appendix provides some necessary background from elementary functional analysis. Audience: The book will be a valuable resource for students and researchers in applied mathematics, physics, mechanics and engineering. The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the notions of optimization and variational methods are employed.
| Author | : Radoslaw Pytlak |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 493 |
| Release | : 2008-11-18 |
| Genre | : Mathematics |
| ISBN | : 354085634X |
This book details algorithms for large-scale unconstrained and bound constrained optimization. It shows optimization techniques from a conjugate gradient algorithm perspective as well as methods of shortest residuals, which have been developed by the author.
| Author | : Roman G. Strongin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 717 |
| Release | : 2013-11-09 |
| Genre | : Mathematics |
| ISBN | : 146154677X |
Everything should be made as simple as possible, but not simpler. (Albert Einstein, Readers Digest, 1977) The modern practice of creating technical systems and technological processes of high effi.ciency besides the employment of new principles, new materials, new physical effects and other new solutions ( which is very traditional and plays the key role in the selection of the general structure of the object to be designed) also includes the choice of the best combination for the set of parameters (geometrical sizes, electrical and strength characteristics, etc.) concretizing this general structure, because the Variation of these parameters ( with the structure or linkage being already set defined) can essentially affect the objective performance indexes. The mathematical tools for choosing these best combinations are exactly what is this book about. With the advent of computers and the computer-aided design the pro bations of the selected variants are usually performed not for the real examples ( this may require some very expensive building of sample op tions and of the special installations to test them ), but by the analysis of the corresponding mathematical models. The sophistication of the mathematical models for the objects to be designed, which is the natu ral consequence of the raising complexity of these objects, greatly com plicates the objective performance analysis. Today, the main (and very often the only) available instrument for such an analysis is computer aided simulation of an object's behavior, based on numerical experiments with its mathematical model.
| Author | : Vladimir F. Demyanov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 420 |
| Release | : 2000-05-31 |
| Genre | : Technology & Engineering |
| ISBN | : 9780792362845 |
2 Radiant sets 236 3 Co-radiant sets 239 4 Radiative and co-radiative sets 241 5 Radiant sets with Lipschitz continuous Minkowski gauges 245 6 Star-shaped sets and their kernels 249 7 Separation 251 8 Abstract convex star-shaped sets 255 References 260 11 DIFFERENCES OF CONVEX COMPACTA AND METRIC SPACES OF CON- 263 VEX COMPACTA WITH APPLICATIONS: A SURVEY A. M. Rubinov, A. A. Vladimirov 1 Introduction 264 2 Preliminaries 264 3 Differences of convex compact sets: general approach 266 4 Metric projections and corresponding differences (one-dimensional case) 267 5 The *-difference 269 6 The Demyanov difference 271 7 Geometric and inductive definitions of the D-difference 273 8 Applications to DC and quasidifferentiable functions 276 9 Differences of pairs of set-valued mappings with applications to quasidiff- entiability 278 10 Applications to approximate subdifferentials 280 11 Applications to the approximation of linear set-valued mappings 281 12 The Demyanov metric 282 13 The Bartels-Pallaschke metric 284 14 Hierarchy of the three norms on Qn 285 15 Derivatives 287 16 Distances from convex polyhedra and convergence of convex polyhedra 289 17 Normality of convex sets 290 18 D-regular sets 291 19 Variable D-regular sets 292 20 Optimization 293 References 294 12 CONVEX APPROXIMATORS.
| Author | : Diethard Klatte |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 351 |
| Release | : 2005-12-17 |
| Genre | : Mathematics |
| ISBN | : 0306476169 |
Many questions dealing with solvability, stability and solution methods for va- ational inequalities or equilibrium, optimization and complementarity problems lead to the analysis of certain (perturbed) equations. This often requires a - formulation of the initial model being under consideration. Due to the specific of the original problem, the resulting equation is usually either not differ- tiable (even if the data of the original model are smooth), or it does not satisfy the assumptions of the classical implicit function theorem. This phenomenon is the main reason why a considerable analytical inst- ment dealing with generalized equations (i.e., with finding zeros of multivalued mappings) and nonsmooth equations (i.e., the defining functions are not c- tinuously differentiable) has been developed during the last 20 years, and that under very different viewpoints and assumptions. In this theory, the classical hypotheses of convex analysis, in particular, monotonicity and convexity, have been weakened or dropped, and the scope of possible applications seems to be quite large. Briefly, this discipline is often called nonsmooth analysis, sometimes also variational analysis. Our book fits into this discipline, however, our main intention is to develop the analytical theory in close connection with the needs of applications in optimization and related subjects. Main Topics of the Book 1. Extended analysis of Lipschitz functions and their generalized derivatives, including ”Newton maps” and regularity of multivalued mappings. 2. Principle of successive approximation under metric regularity and its - plication to implicit functions.
| Author | : Stephen P. Boyd |
| Publisher | : Cambridge University Press |
| Total Pages | : 744 |
| Release | : 2004-03-08 |
| Genre | : Business & Economics |
| ISBN | : 9780521833783 |
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.
