Download or read book Theory and Methods of Vector Optimization Volume Two written by Yu. K. Mashunin and published by Cambridge Scholars Publishing. This book was released on 2021-09-30 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second volume presents research in the field of the mathematical model operation of economic systems, again using as a basis the theory and methods of vector optimization. This volume includes three chapters. The first chapter deals with issues related to the theory of the company, modeling and decision-making, while the second deals with issues related to modeling and decision-making in market systems. The third chapter deals with issues related to modeling, forecasting and decision-making.
Download or read book Theory and Methods of Vector Optimization Volume One written by Yu. K. Mashunin and published by Cambridge Scholars Publishing. This book was released on 2020-03-24 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: This first volume presents the theory and methods of solving vector optimization problems, using initial definitions that include axioms and the optimality principle. This book proves, mathematically, that the result it presents for the solution of the vector (multi-criteria) problem is the optimal outcome, and, as such, solves the problem of vector optimization for the first time. It shows that applied methods of solving vector optimization problems can be used by researchers in modeling and simulating the development of economic systems and technical (engineering) systems.
Download or read book Theory of Vector Optimization written by Dinh The Luc and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes grew out of a series of lectures given by the author at the Univer sity of Budapest during 1985-1986. Additional results have been included which were obtained while the author was at the University of Erlangen-Niirnberg under a grant of the Alexander von Humboldt Foundation. Vector optimization has two main sources coming from economic equilibrium and welfare theories of Edgeworth (1881) and Pareto (1906) and from mathemat ical backgrounds of ordered spaces of Cantor (1897) and Hausdorff (1906). Later, game theory of Borel (1921) and von Neumann (1926) and production theory of Koopmans (1951) have also contributed to this area. However, only in the fifties, after the publication of Kuhn-Tucker's paper (1951) on the necessary and sufficient conditions for efficiency, and of Deubreu's paper (1954) on valuation equilibrium and Pareto optimum, has vector optimization been recognized as a mathematical discipline. The stretching development of this field began later in the seventies and eighties. Today there are a number of books on vector optimization. Most of them are concerned with the methodology and the applications. Few of them offer a systematic study of the theoretical aspects. The aim of these notes is to pro vide a unified background of vector optimization,with the emphasis on nonconvex problems in infinite dimensional spaces ordered by convex cones. The notes are arranged into six chapters. The first chapter presents prelim inary material.
Download or read book Optimization by Vector Space Methods written by David G. Luenberger and published by John Wiley & Sons. This book was released on 1997-01-23 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.
Download or read book Vector Optimization written by Johannes Jahn and published by Springer Science & Business Media. This book was released on 2013-06-05 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: In vector optimization one investigates optimal elements such as min imal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The prob lem of determining at least one of these optimal elements, if they exist at all, is also called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineer ing and economics. Vector optimization problems arise, for exam ple, in functional analysis (the Hahn-Banach theorem, the lemma of Bishop-Phelps, Ekeland's variational principle), multiobjective pro gramming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems). In the last decade vector optimization has been extended to problems with set-valued maps. This new field of research, called set optimiza tion, seems to have important applications to variational inequalities and optimization problems with multivalued data. The roots of vector optimization go back to F. Y. Edgeworth (1881) and V. Pareto (1896) who has already given the definition of the standard optimality concept in multiobjective optimization. But in mathematics this branch of optimization has started with the leg endary paper of H. W. Kuhn and A. W. Tucker (1951). Since about v Vl Preface the end of the 60's research is intensively made in vector optimization.
Download or read book Optimization by Vector Space Methods written by David G. Luenberger and published by John Wiley & Sons. This book was released on 1997-01-23 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.
Download or read book Vector Variational Inequalities and Vector Optimization written by Qamrul Hasan Ansari and published by Springer. This book was released on 2017-10-31 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the mathematical theory of vector variational inequalities and their relations with vector optimization problems. It is the first-ever book to introduce well-posedness and sensitivity analysis for vector equilibrium problems. The first chapter provides basic notations and results from the areas of convex analysis, functional analysis, set-valued analysis and fixed-point theory for set-valued maps, as well as a brief introduction to variational inequalities and equilibrium problems. Chapter 2 presents an overview of analysis over cones, including continuity and convexity of vector-valued functions. The book then shifts its focus to solution concepts and classical methods in vector optimization. It describes the formulation of vector variational inequalities and their applications to vector optimization, followed by separate chapters on linear scalarization, nonsmooth and generalized vector variational inequalities. Lastly, the book introduces readers to vector equilibrium problems and generalized vector equilibrium problems. Written in an illustrative and reader-friendly way, the book offers a valuable resource for all researchers whose work involves optimization and vector optimization.
Download or read book Vector Variational Inequalities and Vector Equilibria written by F. Giannessi and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 522 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book deals with the mathematical theory of vector variational inequalities with special reference to equilibrium problems. Such models have been introduced recently to study new problems from mechanics, structural engineering, networks, and industrial management, and to revisit old ones. The common feature of these problems is that given by the presence of concurrent objectives and by the difficulty of identifying a global functional (like energy) to be extremized. The vector variational inequalities have the advantage of both the variational ones and vector optimization which are found as special cases. Among several applications, the equilibrium flows on a network receive special attention. Audience: The book is addressed to academic researchers as well as industrial ones, in the fields of mathematics, engineering, mathematical programming, control theory, operations research, computer science, and economics.
Download or read book Linear Algebra And Optimization With Applications To Machine Learning Volume Ii Fundamentals Of Optimization Theory With Applications To Machine Learning written by Quaintance Jocelyn and published by World Scientific. This book was released on 2020-03-16 with total page 896 pages. Available in PDF, EPUB and Kindle. Book excerpt: Volume 2 applies the linear algebra concepts presented in Volume 1 to optimization problems which frequently occur throughout machine learning. This book blends theory with practice by not only carefully discussing the mathematical under pinnings of each optimization technique but by applying these techniques to linear programming, support vector machines (SVM), principal component analysis (PCA), and ridge regression. Volume 2 begins by discussing preliminary concepts of optimization theory such as metric spaces, derivatives, and the Lagrange multiplier technique for finding extrema of real valued functions. The focus then shifts to the special case of optimizing a linear function over a region determined by affine constraints, namely linear programming. Highlights include careful derivations and applications of the simplex algorithm, the dual-simplex algorithm, and the primal-dual algorithm. The theoretical heart of this book is the mathematically rigorous presentation of various nonlinear optimization methods, including but not limited to gradient decent, the Karush-Kuhn-Tucker (KKT) conditions, Lagrangian duality, alternating direction method of multipliers (ADMM), and the kernel method. These methods are carefully applied to hard margin SVM, soft margin SVM, kernel PCA, ridge regression, lasso regression, and elastic-net regression. Matlab programs implementing these methods are included.
Download or read book H infinity Engineering and Amplifier Optimization written by Jefferey C. Allen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: H-infinity engineering continues to establish itself as a discipline of applied mathematics. As such, this extensively illustrated monograph makes a significant application of H-infinity theory to electronic amplifier design, demonstrating how recent developments in H-infinity engineering equip amplifier designers with new tools and avenues for research. The presentation, at the interface of applied mathematics and engineering, emphasizes how to (1) compute the best possible performance available from any matching circuits; (2) benchmark existing matching solutions; and (3) generalize results to multiple amplifiers. As the monograph develops, many research directions are pointed out for both disciplines. The physical meaning of a mathematical problem is made explicit for the mathematician, while circuit problems are presented in the H-infinity framework for the engineer. A final chapter organizes these research topics into a collection of open problems ranging from electrical engineering, numerical implementations, and generalizations to H-infinity theory.
Download or read book Recent Developments in Vector Optimization written by Qamrul Hasan Ansari and published by Springer Science & Business Media. This book was released on 2011-09-21 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt: We always come cross several decision-making problems in our daily life. Such problems are always conflicting in which many different view points should be satisfied. In politics, business, industrial systems, management science, networks, etc. one often encounters such kind of problems. The most important and difficult part in such problems is the conflict between various objectives and goals. In these problems, one has to find the minimum(or maximum) for several objective functions. Such problems are called vector optimization problems (VOP),multi-criteria optimization problems or multi-objective optimization problems. This volume deals with several different topics / aspects of vector optimization theory ranging from the very beginning to the most recent one. It contains fourteen chapters written by different experts in the field of vector optimization.
Download or read book Vector Optimization with Infimum and Supremum written by Andreas Löhne and published by Springer Science & Business Media. This book was released on 2011-05-25 with total page 211 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of Vector Optimization is developed by a systematic usage of infimum and supremum. In order to get existence and appropriate properties of the infimum, the image space of the vector optimization problem is embedded into a larger space, which is a subset of the power set, in fact, the space of self-infimal sets. Based on this idea we establish solution concepts, existence and duality results and algorithms for the linear case. The main advantage of this approach is the high degree of analogy to corresponding results of Scalar Optimization. The concepts and results are used to explain and to improve practically relevant algorithms for linear vector optimization problems.
Download or read book OPTIMIZATION AND OPERATIONS RESEARCH Volume II written by Ulrich Derigs and published by EOLSS Publications. This book was released on 2009-02-09 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization and Operations Research is a component of Encyclopedia of Mathematical Sciences in the global Encyclopedia of Life Support Systems (EOLSS), which is an integrated compendium of twenty one Encyclopedias. The Theme on Optimization and Operations Research is organized into six different topics which represent the main scientific areas of the theme: 1. Fundamentals of Operations Research; 2. Advanced Deterministic Operations Research; 3. Optimization in Infinite Dimensions; 4. Game Theory; 5. Stochastic Operations Research; 6. Decision Analysis, which are then expanded into multiple subtopics, each as a chapter. These four volumes are aimed at the following five major target audiences: University and College students Educators, Professional Practitioners, Research Personnel and Policy Analysts, Managers, and Decision Makers and NGOs.
Download or read book Convex Optimization with Computational Errors written by Alexander J. Zaslavski and published by Springer Nature. This book was released on 2020-01-31 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to the study of approximate solutions of optimization problems in the presence of computational errors. It contains a number of results on the convergence behavior of algorithms in a Hilbert space, which are known as important tools for solving optimization problems. The research presented in the book is the continuation and the further development of the author's (c) 2016 book Numerical Optimization with Computational Errors, Springer 2016. Both books study the algorithms taking into account computational errors which are always present in practice. The main goal is, for a known computational error, to find out what an approximate solution can be obtained and how many iterates one needs for this. The main difference between this new book and the 2016 book is that in this present book the discussion takes into consideration the fact that for every algorithm, its iteration consists of several steps and that computational errors for different steps are generally, different. This fact, which was not taken into account in the previous book, is indeed important in practice. For example, the subgradient projection algorithm consists of two steps. The first step is a calculation of a subgradient of the objective function while in the second one we calculate a projection on the feasible set. In each of these two steps there is a computational error and these two computational errors are different in general. It may happen that the feasible set is simple and the objective function is complicated. As a result, the computational error, made when one calculates the projection, is essentially smaller than the computational error of the calculation of the subgradient. Clearly, an opposite case is possible too. Another feature of this book is a study of a number of important algorithms which appeared recently in the literature and which are not discussed in the previous book. This monograph contains 12 chapters. Chapter 1 is an introduction. In Chapter 2 we study the subgradient projection algorithm for minimization of convex and nonsmooth functions. We generalize the results of [NOCE] and establish results which has no prototype in [NOCE]. In Chapter 3 we analyze the mirror descent algorithm for minimization of convex and nonsmooth functions, under the presence of computational errors. For this algorithm each iteration consists of two steps. The first step is a calculation of a subgradient of the objective function while in the second one we solve an auxiliary minimization problem on the set of feasible points. In each of these two steps there is a computational error. We generalize the results of [NOCE] and establish results which has no prototype in [NOCE]. In Chapter 4 we analyze the projected gradient algorithm with a smooth objective function under the presence of computational errors. In Chapter 5 we consider an algorithm, which is an extension of the projection gradient algorithm used for solving linear inverse problems arising in signal/image processing. In Chapter 6 we study continuous subgradient method and continuous subgradient projection algorithm for minimization of convex nonsmooth functions and for computing the saddle points of convex-concave functions, under the presence of computational errors. All the results of this chapter has no prototype in [NOCE]. In Chapters 7-12 we analyze several algorithms under the presence of computational errors which were not considered in [NOCE]. Again, each step of an iteration has a computational errors and we take into account that these errors are, in general, different. An optimization problems with a composite objective function is studied in Chapter 7. A zero-sum game with two-players is considered in Chapter 8. A predicted decrease approximation-based method is used in Chapter 9 for constrained convex optimization. Chapter 10 is devoted to minimization of quasiconvex functions. Minimization of sharp weakly convex functions is discussed in Chapter 11. Chapter 12 is devoted to a generalized projected subgradient method for minimization of a convex function over a set which is not necessarily convex. The book is of interest for researchers and engineers working in optimization. It also can be useful in preparation courses for graduate students. The main feature of the book which appeals specifically to this audience is the study of the influence of computational errors for several important optimization algorithms. The book is of interest for experts in applications of optimization to engineering and economics.
Download or read book Structural Optimization written by A. Borkowski and published by Springer Science & Business Media. This book was released on 1990-01-31 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Vector Optimization and Monotone Operators via Convex Duality written by Sorin-Mihai Grad and published by Springer. This book was released on 2014-09-03 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book investigates several duality approaches for vector optimization problems, while also comparing them. Special attention is paid to duality for linear vector optimization problems, for which a vector dual that avoids the shortcomings of the classical ones is proposed. Moreover, the book addresses different efficiency concepts for vector optimization problems. Among the problems that appear when the framework is generalized by considering set-valued functions, an increasing interest is generated by those involving monotone operators, especially now that new methods for approaching them by means of convex analysis have been developed. Following this path, the book provides several results on different properties of sums of monotone operators.
Download or read book Support Vector Machines written by Naiyang Deng and published by CRC Press. This book was released on 2012-12-17 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: Support Vector Machines: Optimization Based Theory, Algorithms, and Extensions presents an accessible treatment of the two main components of support vector machines (SVMs)-classification problems and regression problems. The book emphasizes the close connection between optimization theory and SVMs since optimization is one of the pillars on which
