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Author: Shashi K. Mishra Publisher: Springer Science & Business Media ISBN: 1441996400 Category : Business & Economics Languages : en Pages : 276
Book Description
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: Shashi K. Mishra Publisher: Springer Science & Business Media ISBN: 1441996400 Category : Business & Economics Languages : en Pages : 276
Book Description
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: Shashi K. Mishra Publisher: Springer ISBN: 9781441996398 Category : Business & Economics Languages : en Pages : 270
Book Description
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: Ying Cui Publisher: Society for Industrial and Applied Mathematics (SIAM) ISBN: 9781611976731 Category : Convex functions Languages : en Pages : 0
Book Description
"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: Prateek Jain Publisher: Foundations and Trends in Machine Learning ISBN: 9781680833683 Category : Machine learning Languages : en Pages : 218
Book Description
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: Bingyuan Liu Publisher: ISBN: Category : Languages : en Pages :
Book Description
Many machine learning models need to solve nonconvex and nonsmooth optimization problems. Compared with convex optimization, nonconvex optimization captures the intrinsic structure of the learning problem more accurately. But, there are usually no well-developed algorithms with convergence guarantees for solving nonconvex and nonsmooth optimization problems. This thesis investigates how to design efficient algorithms with convergence guarantees and establish statistical properties for the computed solutions in these nonconvex learning problems. In the first part of this thesis, we study three nonconvex high-dimensional statistical learning problems. In chapter 3, we propose a robust high-dimensional regression estimator with coefficient thresholding. The coefficient thresholding is imposed in the loss function to handle the strong dependence between predictors but leads to a nonconvex loss function. We propose an efficient composite gradient descent algorithm to solve the optimization with convergence guarantee and prove the estimation consistency of our proposed estimator. In chapter 4, we propose a sparse estimation of semiparametric covariate-adjusted graphical models. In chapter 5, we study sparse sufficient dimension reduction estimators. We study the theoretical property of nonconvex penalize estimators for both chapters and propose nonconvex ADMM algorithms to solve them with computational guarantees efficiently. In the second part of this thesis, we study nonconvex neural network models. First, we study the loss landscape of attention mechanisms, which is a widely used module in deep learning. Theoretically and empirically, we show that neural network models with attention mechanisms have lower sample complexity, better generalization, and maintain a good loss landscape structure. Second, we propose a novel neural network layer that improved model robustness against adversarial attacks through neighborhood preservation. We show that despite a highly nonconvex nature, our layer has a lower Lipschitz bound, thus more robust against adversarial attacks.
Author: Roman G. Strongin Publisher: Springer Science & Business Media ISBN: 146154677X Category : Mathematics Languages : en Pages : 717
Book Description
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: David Yang Gao Publisher: Springer Science & Business Media ISBN: 9780792361459 Category : Mathematics Languages : en Pages : 476
Book Description
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 ISBN: 354085634X Category : Mathematics Languages : en Pages : 493
Book Description
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: F.H. Clarke Publisher: Springer Science & Business Media ISBN: 1475760191 Category : Science Languages : en Pages : 481
Book Description
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: Shashi K. Mishra
Author: Shashi K. Mishra
Author: Shashi Kant Mishra
Author: Shashi K. Mishra
Author: Ying Cui
Author: Prateek Jain
Author: Bingyuan Liu
Author: Roman G. Strongin
Author: David Yang Gao
Author: Radoslaw Pytlak
Author: F.H. Clarke