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Nonsmooth Equations In Optimization
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Book Synopsis Nonsmooth Equations in Optimization by : Diethard Klatte
Download or read book Nonsmooth Equations in Optimization written by Diethard Klatte and published by Springer Science & Business Media. This book was released on 2005-12-17 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Nonsmooth Optimization: Analysis And Algorithms With Applications To Optimal Control by : Marko M Makela
Download or read book Nonsmooth Optimization: Analysis And Algorithms With Applications To Optimal Control written by Marko M Makela and published by World Scientific. This book was released on 1992-05-07 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a self-contained elementary study for nonsmooth analysis and optimization, and their use in solution of nonsmooth optimal control problems. The first part of the book is concerned with nonsmooth differential calculus containing necessary tools for nonsmooth optimization. The second part is devoted to the methods of nonsmooth optimization and their development. A proximal bundle method for nonsmooth nonconvex optimization subject to nonsmooth constraints is constructed. In the last part nonsmooth optimization is applied to problems arising from optimal control of systems covered by partial differential equations. Several practical problems, like process control and optimal shape design problems are considered.
Book Synopsis Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods by : Masao Fukushima
Download or read book Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods written by Masao Fukushima and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: The concept of "reformulation" has long been playing an important role in mathematical programming. A classical example is the penalization technique in constrained optimization that transforms the constraints into the objective function via a penalty function thereby reformulating a constrained problem as an equivalent or approximately equivalent unconstrained problem. More recent trends consist of the reformulation of various mathematical programming prob lems, including variational inequalities and complementarity problems, into equivalent systems of possibly nonsmooth, piecewise smooth or semismooth nonlinear equations, or equivalent unconstrained optimization problems that are usually differentiable, but in general not twice differentiable. Because of the recent advent of various tools in nonsmooth analysis, the reformulation approach has become increasingly profound and diversified. In view of growing interests in this active field, we planned to organize a cluster of sessions entitled "Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods" in the 16th International Symposium on Mathematical Programming (ismp97) held at Lausanne EPFL, Switzerland on August 24-29, 1997. Responding to our invitation, thirty-eight people agreed to give a talk within the cluster, which enabled us to organize thirteen sessions in total. We think that it was one of the largest and most exciting clusters in the symposium. Thanks to the earnest support by the speakers and the chairpersons, the sessions attracted much attention of the participants and were filled with great enthusiasm of the audience.
Book Synopsis Nonsmooth Vector Functions and Continuous Optimization by : V. Jeyakumar
Download or read book Nonsmooth Vector Functions and Continuous Optimization written by V. Jeyakumar and published by Springer Science & Business Media. This book was released on 2007-10-23 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems, as well as variational inequalities in finite dimensions. The treatment is motivated by a desire to expose an elementary approach to nonsmooth calculus, using a set of matrices to replace the nonexistent Jacobian matrix of a continuous vector function.
Book Synopsis Nonsmooth Approach to Optimization Problems with Equilibrium Constraints by : Jiri Outrata
Download or read book Nonsmooth Approach to Optimization Problems with Equilibrium Constraints written by Jiri Outrata and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the early fifties, applied mathematicians, engineers and economists started to pay c10se attention to the optimization problems in which another (lower-Ievel) optimization problem arises as a side constraint. One of the motivating factors was the concept of the Stackelberg solution in game theory, together with its economic applications. Other problems have been encountered in the seventies in natural sciences and engineering. Many of them are of practical importance and have been extensively studied, mainly from the theoretical point of view. Later, applications to mechanics and network design have lead to an extension of the problem formulation: Constraints in form of variation al inequalities and complementarity problems were also admitted. The term "generalized bi level programming problems" was used at first but later, probably in Harker and Pang, 1988, a different terminology was introduced: Mathematical programs with equilibrium constraints, or simply, MPECs. In this book we adhere to MPEC terminology. A large number of papers deals with MPECs but, to our knowledge, there is only one monograph (Luo et al. , 1997). This monograph concentrates on optimality conditions and numerical methods. Our book is oriented similarly, but we focus on those MPECs which can be treated by the implicit programming approach: the equilibrium constraint locally defines a certain implicit function and allows to convert the problem into a mathematical program with a nonsmooth objective.
Book Synopsis Recent Advances in Nonsmooth Optimization by : Dingzhu Du
Download or read book Recent Advances in Nonsmooth Optimization written by Dingzhu Du and published by World Scientific. This book was released on 1995 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonsmooth optimization covers the minimization or maximization of functions which do not have the differentiability properties required by classical methods. The field of nonsmooth optimization is significant, not only because of the existence of nondifferentiable functions arising directly in applications, but also because several important methods for solving difficult smooth problems lead directly to the need to solve nonsmooth problems, which are either smaller in dimension or simpler in structure.This book contains twenty five papers written by forty six authors from twenty countries in five continents. It includes papers on theory, algorithms and applications for problems with first-order nondifferentiability (the usual sense of nonsmooth optimization) second-order nondifferentiability, nonsmooth equations, nonsmooth variational inequalities and other problems related to nonsmooth optimization.
Book Synopsis Methods of Dynamic and Nonsmooth Optimization by : Frank H. Clarke
Download or read book Methods of Dynamic and Nonsmooth Optimization written by Frank H. Clarke and published by SIAM. This book was released on 1989-01-01 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents the elements of a unified approach to optimization based on 'nonsmooth analysis', a term introduced in the 1970's by the author, who is a pioneer in the field. Based on a series of lectures given at a conference at Emory University in 1986, this volume presents its subjects in a self-contained and accessible manner. The topics treated here have been in an active state of development. Focuses mainly on deterministic optimal control, the calculus of variations, and mathematical programming. In addition, it features a tutorial in nonsmooth analysis and geometry and demonstrates that the method of value function analysis via proximal normals is a powerful tool in the study of necessary conditions, sufficient conditions, controllability, and sensitivity analysis. The distinction between inductive and deductive methods, the use of Hamiltonians, the verification technique, and penalization are also emphasized.
Book Synopsis Optimization and Nonsmooth Analysis by : Frank H. Clarke
Download or read book Optimization and Nonsmooth Analysis written by Frank H. Clarke and published by SIAM. This book was released on 1990-01-01 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical Reviews said of this book that it was 'destined to become a classical reference.' This book has appeared in Russian translation and has been praised both for its lively exposition and its fundamental contributions. The author first develops a general theory of nonsmooth analysis and geometry which, together with a set of associated techniques, has had a profound effect on several branches of analysis and optimization. Clarke then applies these methods to obtain a powerful, unified approach to the analysis of problems in optimal control and mathematical programming. Examples are drawn from economics, engineering, mathematical physics, and various branches of analysis in this reprint volume.
Book Synopsis Nonsmooth Analysis and Control Theory by : Francis H. Clarke
Download or read book Nonsmooth Analysis and Control Theory written by Francis H. Clarke and published by Springer Science & Business Media. This book was released on 2008-01-10 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: A clear and succinct presentation of the essentials of this subject, together with some of its applications and a generous helping of interesting exercises. Following an introductory chapter with a taste of what is to come, the next three chapters constitute a course in nonsmooth analysis and identify a coherent and comprehensive approach to the subject, leading to an efficient, natural, and powerful body of theory. The whole is rounded off with a self-contained introduction to the theory of control of ordinary differential equations. The authors have incorporated a number of new results which clarify the relationships between the different schools of thought in the subject, with the aim of making nonsmooth analysis accessible to a wider audience. End-of-chapter problems offer scope for deeper understanding.
Book Synopsis Generalized Newton-type Methods for Nonsmooth Equations in Optimization and Complementarity Problems by : Stephan Bütikofer
Download or read book Generalized Newton-type Methods for Nonsmooth Equations in Optimization and Complementarity Problems written by Stephan Bütikofer and published by . This book was released on 2008 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Introduction to Functional Analysis by : Christian Clason
Download or read book Introduction to Functional Analysis written by Christian Clason and published by Springer Nature. This book was released on 2020-11-30 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: Functional analysis has become one of the essential foundations of modern applied mathematics in the last decades, from the theory and numerical solution of differential equations, from optimization and probability theory to medical imaging and mathematical image processing. This textbook offers a compact introduction to the theory and is designed to be used during one semester, fitting exactly 26 lectures of 90 minutes each. It ranges from the topological fundamentals recalled from basic lectures on real analysis to spectral theory in Hilbert spaces. Special attention is given to the central results on dual spaces and weak convergence.
Book Synopsis Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models by : F. Giannessi
Download or read book Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models written by F. Giannessi and published by Springer Science & Business Media. This book was released on 2006-04-11 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the book is to cover the three fundamental aspects of research in equilibrium problems: the statement problem and its formulation using mainly variational methods, its theoretical solution by means of classical and new variational tools, the calculus of solutions and applications in concrete cases. The book shows how many equilibrium problems follow a general law (the so-called user equilibrium condition). Such law allows us to express the problem in terms of variational inequalities. Variational inequalities provide a powerful methodology, by which existence and calculation of the solution can be obtained.
Book Synopsis Nonsmooth Optimization and Related Topics by : F.H. Clarke
Download or read book Nonsmooth Optimization and Related Topics written by F.H. Clarke and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization by : Qamrul Hasan Ansari
Download or read book Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization written by Qamrul Hasan Ansari and published by CRC Press. This book was released on 2013-07-18 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: Until now, no book addressed convexity, monotonicity, and variational inequalities together. Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization covers all three topics, including new variational inequality problems defined by a bifunction.The first part of the book focuses on generalized convexity and generalized
Book Synopsis Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces by : Michael Ulbrich
Download or read book Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces written by Michael Ulbrich and published by SIAM. This book was released on 2011-07-28 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive treatment of semismooth Newton methods in function spaces: from their foundations to recent progress in the field. This book is appropriate for researchers and practitioners in PDE-constrained optimization, nonlinear optimization and numerical analysis, as well as engineers interested in the current theory and methods for solving variational inequalities.
Book Synopsis Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization by : Qamrul Hasan Ansari
Download or read book Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization written by Qamrul Hasan Ansari and published by CRC Press. This book was released on 2013-07-18 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: Until now, no book addressed convexity, monotonicity, and variational inequalities together. Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization covers all three topics, including new variational inequality problems defined by a bifunction. The first part of the book focuses on generalized convexity and generalized monotonicity. The authors investigate convexity and generalized convexity for both the differentiable and nondifferentiable case. For the nondifferentiable case, they introduce the concepts in terms of a bifunction and the Clarke subdifferential. The second part offers insight into variational inequalities and optimization problems in smooth as well as nonsmooth settings. The book discusses existence and uniqueness criteria for a variational inequality, the gap function associated with it, and numerical methods to solve it. It also examines characterizations of a solution set of an optimization problem and explores variational inequalities defined by a bifunction and set-valued version given in terms of the Clarke subdifferential. Integrating results on convexity, monotonicity, and variational inequalities into one unified source, this book deepens your understanding of various classes of problems, such as systems of nonlinear equations, optimization problems, complementarity problems, and fixed-point problems. The book shows how variational inequality theory not only serves as a tool for formulating a variety of equilibrium problems, but also provides algorithms for computational purposes.
Book Synopsis Nonsmooth Optimization in Honor of the 60th Birthday of Adil M. Bagirov by : Napsu Karmitsa
Download or read book Nonsmooth Optimization in Honor of the 60th Birthday of Adil M. Bagirov written by Napsu Karmitsa and published by MDPI. This book was released on 2020-12-18 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book was to collect the most recent methods developed for NSO and its practical applications. The book contains seven papers: The first is the foreword by the Guest Editors giving a brief review of NSO and its real-life applications and acknowledging the outstanding contributions of Professor Adil Bagirov to both the theoretical and practical aspects of NSO. The second paper introduces a new and very efficient algorithm for solving uncertain unit-commitment (UC) problems. The third paper proposes a new nonsmooth version of the generalized damped Gauss–Newton method for solving nonlinear complementarity problems. In the fourth paper, the abs-linear representation of piecewise linear functions is extended to yield simultaneously their DC decomposition as well as the pair of generalized gradients. The fifth paper presents the use of biased-randomized algorithms as an effective methodology to cope with NP-hard and nonsmooth optimization problems in many practical applications. In the sixth paper, a problem concerning the scheduling of nuclear waste disposal is modeled as a nonsmooth multiobjective mixed-integer nonlinear optimization problem, and a novel method using the two-slope parameterized achievement scalarizing functions is introduced. Finally, the last paper considers binary classification of a multiple instance learning problem and formulates the learning problem as a nonconvex nonsmooth unconstrained optimization problem with a DC objective function.
