Nonsmooth Approach To Optimization Problems With Equilibrium Constraints

Author : Jiri Outrata,M. Kocvara,J. Zowe
Publisher : Springer Science & Business Media
Page : 281 pages
File Size : 49,8 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475728255

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Nonsmooth Approach to Optimization Problems with Equilibrium Constraints by Jiri Outrata,M. Kocvara,J. Zowe Pdf

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.

Author : Michal Kočvara,Jiří V. Outrata
Publisher : Unknown
Page : 34 pages
File Size : 45,6 Mb
Release : 1996
Category : Electronic
ISBN : OCLC:75974194

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A Nonsmooth Approach to Optimization Problems with Equilibrium Constraints by Michal Kočvara,Jiří V. Outrata Pdf

Author : F. Giannessi,A. Maugeri,Panos M. Pardalos
Publisher : Springer Science & Business Media
Page : 304 pages
File Size : 43,7 Mb
Release : 2006-04-11
Category : Mathematics
ISBN : 9780306480263

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Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models by F. Giannessi,A. Maugeri,Panos M. Pardalos Pdf

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.

Author : Masao Fukushima,Liqun Qi
Publisher : Springer Science & Business Media
Page : 440 pages
File Size : 44,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475763881

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Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods by Masao Fukushima,Liqun Qi Pdf

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.

Author : Zhi-Quan Luo,Jong-Shi Pang,Daniel Ralph
Publisher : Cambridge University Press
Page : 128 pages
File Size : 54,7 Mb
Release : 1996-11-13
Category : Mathematics
ISBN : 9781316582619

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Mathematical Programs with Equilibrium Constraints by Zhi-Quan Luo,Jong-Shi Pang,Daniel Ralph Pdf

This book provides a solid foundation and an extensive study for an important class of constrained optimization problems known as Mathematical Programs with Equilibrium Constraints (MPEC), which are extensions of bilevel optimization problems. The book begins with the description of many source problems arising from engineering and economics that are amenable to treatment by the MPEC methodology. Error bounds and parametric analysis are the main tools to establish a theory of exact penalisation, a set of MPEC constraint qualifications and the first-order and second-order optimality conditions. The book also describes several iterative algorithms such as a penalty-based interior point algorithm, an implicit programming algorithm and a piecewise sequential quadratic programming algorithm for MPECs. Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.

Author : Vivek Laha,Pierre Maréchal,S. K. Mishra
Publisher : Springer Nature
Page : 441 pages
File Size : 49,9 Mb
Release : 2021-07-27
Category : Mathematics
ISBN : 9789811618192

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Optimization, Variational Analysis and Applications by Vivek Laha,Pierre Maréchal,S. K. Mishra Pdf

This book includes selected papers presented at the Indo-French Seminar on Optimization, Variational Analysis and Applications (IFSOVAA-2020), held at the Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India, from 2–4 February 2020. The book discusses current optimization problems and their solutions by using the powerful tool of variational analysis. Topics covered in this volume include set optimization, multiobjective optimization, mathematical programs with complementary, equilibrium, vanishing and switching constraints, copositive optimization, interval-valued optimization, sequential quadratic programming, bound-constrained optimization, variational inequalities, and more. Several applications in different branches of applied mathematics, engineering, economics, finance, and medical sciences have been included. Each chapter not only provides a detailed survey of the topic but also builds systematic theories and suitable algorithms to deduce the most recent findings in literature. This volume appeals to graduate students as well as researchers and practitioners in pure and applied mathematics and related fields that make use of variational analysis in solving optimization problems.

Author : Didier Aussel,C.S. Lalitha
Publisher : Springer
Page : 124 pages
File Size : 52,5 Mb
Release : 2018-04-03
Category : Mathematics
ISBN : 9789811047749

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Generalized Nash Equilibrium Problems, Bilevel Programming and MPEC by Didier Aussel,C.S. Lalitha Pdf

The book discusses three classes of problems: the generalized Nash equilibrium problems, the bilevel problems and the mathematical programming with equilibrium constraints (MPEC). These problems interact through their mathematical analysis as well as their applications. The primary aim of the book is to present the modern tool of variational analysis and optimization, which are used to analyze these three classes of problems. All contributing authors are respected academicians, scientists and researchers from around the globe. These contributions are based on the lectures delivered by experts at CIMPA School, held at the University of Delhi, India, from 25 November–6 December 2013, and peer-reviewed by international experts. The book contains five chapters. Chapter 1 deals with nonsmooth, nonconvex bilevel optimization problems whose feasible set is described by using the graph of the solution set mapping of a parametric optimization problem. Chapter 2 describes a constraint qualification to MPECs considered as an application of calmness concept of multifunctions and is used to derive M-stationarity conditions for MPEC. Chapter 3 discusses the first- and second-order optimality conditions derived for a special case of a bilevel optimization problem in which the constraint set of the lower level problem is described as a general compact convex set. Chapter 4 concentrates the results of the modelization and analysis of deregulated electricity markets with a focus on auctions and mechanism design. Chapter 5 focuses on optimization approaches called reflection methods for protein conformation determination within the framework of matrix completion. The last chapter (Chap. 6) deals with the single-valuedness of quasimonotone maps by using the concept of single-directionality with a special focus on the case of the normal operator of lower semi-continuous quasiconvex functions.

Author : Altannar Chinchuluun,Panos M. Pardalos,Rentsen Enkhbat,Efstratios N. Pistikopoulos
Publisher : Springer Science & Business Media
Page : 351 pages
File Size : 47,8 Mb
Release : 2012-11-28
Category : Mathematics
ISBN : 9781461451310

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Optimization, Simulation, and Control by Altannar Chinchuluun,Panos M. Pardalos,Rentsen Enkhbat,Efstratios N. Pistikopoulos Pdf

Optimization, simulation and control play an increasingly important role in science and industry. Because of their numerous applications in various disciplines, research in these areas is accelerating at a rapid pace. This volume brings together the latest developments in these areas of research as well as presents applications of these results to a wide range of real-world problems. The book is composed of invited contributions by experts from around the world who work to develop and apply new optimization, simulation and control techniques either at a theoretical level or in practice. Some key topics presented include: equilibrium problems, multi-objective optimization, variational inequalities, stochastic processes, numerical analysis, optimization in signal processing, and various other interdisciplinary applications. This volume can serve as a useful resource for researchers, practitioners, and advanced graduate students of mathematics and engineering working in research areas where results in optimization, simulation and control can be applied.

Author : Vladimir Shikhman
Publisher : Springer Science & Business Media
Page : 200 pages
File Size : 46,8 Mb
Release : 2011-11-18
Category : Mathematics
ISBN : 9781461418979

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Topological Aspects of Nonsmooth Optimization by Vladimir Shikhman Pdf

This book deals with nonsmooth structures arising within the optimization setting. It considers four optimization problems, namely, mathematical programs with complementarity constraints, general semi-infinite programming problems, mathematical programs with vanishing constraints and bilevel optimization. The author uses the topological approach and topological invariants of corresponding feasible sets are investigated. Moreover, the critical point theory in the sense of Morse is presented and parametric and stability issues are considered. The material progresses systematically and establishes a comprehensive theory for a rather broad class of optimization problems tailored to their particular type of nonsmoothness. Topological Aspects of Nonsmooth Optimization will benefit researchers and graduate students in applied mathematics, especially those working in optimization theory, nonsmooth analysis, algebraic topology and singularity theory. ​ ​

Author : Frank H. Clarke
Publisher : SIAM
Page : 95 pages
File Size : 41,8 Mb
Release : 1989-01-01
Category : Mathematics
ISBN : 1611970148

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Methods of Dynamic and Nonsmooth Optimization by Frank H. Clarke Pdf

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, and this work therefore incorporates more recent results than those presented in 1986. 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.

Author : Ashkan Nikeghbali,Panos M. Pardalos,Andrei M. Raigorodskii,Michael Th. Rassias
Publisher : Springer Nature
Page : 417 pages
File Size : 41,8 Mb
Release : 2022-08-04
Category : Mathematics
ISBN : 9783031008320

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High-Dimensional Optimization and Probability by Ashkan Nikeghbali,Panos M. Pardalos,Andrei M. Raigorodskii,Michael Th. Rassias Pdf

This volume presents extensive research devoted to a broad spectrum of mathematics with emphasis on interdisciplinary aspects of Optimization and Probability. Chapters also emphasize applications to Data Science, a timely field with a high impact in our modern society. The discussion presents modern, state-of-the-art, research results and advances in areas including non-convex optimization, decentralized distributed convex optimization, topics on surrogate-based reduced dimension global optimization in process systems engineering, the projection of a point onto a convex set, optimal sampling for learning sparse approximations in high dimensions, the split feasibility problem, higher order embeddings, codifferentials and quasidifferentials of the expectation of nonsmooth random integrands, adjoint circuit chains associated with a random walk, analysis of the trade-off between sample size and precision in truncated ordinary least squares, spatial deep learning, efficient location-based tracking for IoT devices using compressive sensing and machine learning techniques, and nonsmooth mathematical programs with vanishing constraints in Banach spaces. The book is a valuable source for graduate students as well as researchers working on Optimization, Probability and their various interconnections with a variety of other areas. Chapter 12 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Author : Fausto Pedro García Márquez,Benjamin Lev
Publisher : Springer
Page : 267 pages
File Size : 54,9 Mb
Release : 2016-11-15
Category : Computers
ISBN : 9783319454986

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Big Data Management by Fausto Pedro García Márquez,Benjamin Lev Pdf

This book focuses on the analytic principles of business practice and big data. Specifically, it provides an interface between the main disciplines of engineering/technology and the organizational and administrative aspects of management, serving as a complement to books in other disciplines such as economics, finance, marketing and risk analysis. The contributors present their areas of expertise, together with essential case studies that illustrate the successful application of engineering management theories in real-life examples.

Author : Adil M. Bagirov,Manlio Gaudioso,Napsu Karmitsa,Marko M. Mäkelä,Sona Taheri
Publisher : Springer Nature
Page : 696 pages
File Size : 45,6 Mb
Release : 2020-02-28
Category : Business & Economics
ISBN : 9783030349103

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Numerical Nonsmooth Optimization by Adil M. Bagirov,Manlio Gaudioso,Napsu Karmitsa,Marko M. Mäkelä,Sona Taheri Pdf

Solving nonsmooth optimization (NSO) problems is critical in many practical applications and real-world modeling systems. The aim of this book is to survey various numerical methods for solving NSO problems and to provide an overview of the latest developments in the field. Experts from around the world share their perspectives on specific aspects of numerical NSO. The book is divided into four parts, the first of which considers general methods including subgradient, bundle and gradient sampling methods. In turn, the second focuses on methods that exploit the problem’s special structure, e.g. algorithms for nonsmooth DC programming, VU decomposition techniques, and algorithms for minimax and piecewise differentiable problems. The third part considers methods for special problems like multiobjective and mixed integer NSO, and problems involving inexact data, while the last part highlights the latest advancements in derivative-free NSO. Given its scope, the book is ideal for students attending courses on numerical nonsmooth optimization, for lecturers who teach optimization courses, and for practitioners who apply nonsmooth optimization methods in engineering, artificial intelligence, machine learning, and business. Furthermore, it can serve as a reference text for experts dealing with nonsmooth optimization.

Author : Anurag Jayswal,Tadeusz Antczak
Publisher : CRC Press
Page : 309 pages
File Size : 42,6 Mb
Release : 2022-09-13
Category : Technology & Engineering
ISBN : 9781000648980

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Continuous Optimization and Variational Inequalities by Anurag Jayswal,Tadeusz Antczak Pdf

The proposed book provides a comprehensive coverage of theory and methods in the areas of continuous optimization and variational inequality. It describes theory and solution methods for optimization with smooth and non-smooth functions, for variational inequalities with single-valued and multivalued mappings, and for related classes such as mixed variational inequalities, complementarity problems, and general equilibrium problems. The emphasis is made on revealing generic properties of these problems that allow creation of efficient solution methods. Salient Features The book presents a deep, wide-ranging introduction to the theory of the optimal control of processes governed by optimization techniques and variational inequality Several solution methods are provided which will help the reader to develop various optimization tools for real-life problems which can be modeled by optimization techniques involving linear and nonlinear functions. The book focuses on most recent contributions in the nonlinear phenomena, which can appear in various areas of human activities. This book also presents relevant mathematics clearly and simply to help solve real life problems in diverse fields such as mechanical engineering, management, control behavior, traffic signal, industry, etc. This book is aimed primarily at advanced undergraduates and graduate students pursuing computer engineering and electrical engineering courses. Researchers, academicians and industry people will also find this book useful.

Author : V. Jeyakumar,Dinh The Luc
Publisher : Springer Science & Business Media
Page : 277 pages
File Size : 47,7 Mb
Release : 2007-10-23
Category : Mathematics
ISBN : 9780387737171

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Nonsmooth Vector Functions and Continuous Optimization by V. Jeyakumar,Dinh The Luc Pdf

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.