Quadratic Variational Problems
Author: Jack Indritz
Publisher:
Published: 1957
Total Pages: 60
ISBN-13:
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Quadratic Programming and Affine Variational Inequalities
Author: Gue Myung Lee
Publisher: Springer Science & Business Media
Published: 2006-03-30
Total Pages: 353
ISBN-13: 0387242783
DOWNLOAD EBOOKQuadratic programs and affine variational inequalities represent two fundamental, closely-related classes of problems in the t,heories of mathematical programming and variational inequalities, resp- tively. This book develops a unified theory on qualitative aspects of nonconvex quadratic programming and affine variational inequ- ities. The first seven chapters introduce the reader step-by-step to the central issues concerning a quadratic program or an affine variational inequality, such as the solution existence, necessary and sufficient conditions for a point to belong to the solution set, and properties of the solution set. The subsequent two chapters discuss briefly two concrete nlodels (linear fractional vector optimization and the traffic equilibrium problem) whose analysis can benefit a lot from using the results on quadratic programs and affine variational inequalities. There are six chapters devoted to the study of conti- ity and/or differentiability properties of the characteristic maps and functions in quadratic programs and in affine variational inequa- ties where all the components of the problem data are subject to perturbation. Quadratic programs and affine variational inequa- ties under linear perturbations are studied in three other chapters. One special feature of the presentation is that when a certain pr- erty of a characteristic map or function is investigated, we always try first to establish necessary conditions for it to hold, then we go on to study whether the obtained necessary conditions are suf- cient ones. This helps to clarify the structures of the two classes of problems under consideration.
Newton-Type Methods for Optimization and Variational Problems
Author: Alexey F. Izmailov
Publisher: Springer
Published: 2014-07-08
Total Pages: 587
ISBN-13: 3319042475
DOWNLOAD EBOOKThis book presents comprehensive state-of-the-art theoretical analysis of the fundamental Newtonian and Newtonian-related approaches to solving optimization and variational problems. A central focus is the relationship between the basic Newton scheme for a given problem and algorithms that also enjoy fast local convergence. The authors develop general perturbed Newtonian frameworks that preserve fast convergence and consider specific algorithms as particular cases within those frameworks, i.e., as perturbations of the associated basic Newton iterations. This approach yields a set of tools for the unified treatment of various algorithms, including some not of the Newton type per se. Among the new subjects addressed is the class of degenerate problems. In particular, the phenomenon of attraction of Newton iterates to critical Lagrange multipliers and its consequences as well as stabilized Newton methods for variational problems and stabilized sequential quadratic programming for optimization. This volume will be useful to researchers and graduate students in the fields of optimization and variational analysis.
Nonlinear Analysis and Variational Problems
Author: Panos M. Pardalos
Publisher: Springer Science & Business Media
Published: 2009-10-20
Total Pages: 502
ISBN-13: 1441901582
DOWNLOAD EBOOKThe chapters in this volume, written by international experts from different fields of mathematics, are devoted to honoring George Isac, a renowned mathematician. These contributions focus on recent developments in complementarity theory, variational principles, stability theory of functional equations, nonsmooth optimization, and several other important topics at the forefront of nonlinear analysis and optimization.
Variational Analysis
Author: R. Tyrrell Rockafellar
Publisher: Springer Science & Business Media
Published: 2009-06-26
Total Pages: 747
ISBN-13: 3642024319
DOWNLOAD EBOOKFrom its origins in the minimization of integral functionals, the notion of variations has evolved greatly in connection with applications in optimization, equilibrium, and control. This book develops a unified framework and provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, and normal integrands.
Optimal Quadratic Programming Algorithms
Author: Zdenek Dostál
Publisher: Springer Science & Business Media
Published: 2009-04-03
Total Pages: 293
ISBN-13: 0387848061
DOWNLOAD EBOOKQuadratic programming (QP) is one advanced mathematical technique that allows for the optimization of a quadratic function in several variables in the presence of linear constraints. This book presents recently developed algorithms for solving large QP problems and focuses on algorithms which are, in a sense optimal, i.e., they can solve important classes of problems at a cost proportional to the number of unknowns. For each algorithm presented, the book details its classical predecessor, describes its drawbacks, introduces modifications that improve its performance, and demonstrates these improvements through numerical experiments. This self-contained monograph can serve as an introductory text on quadratic programming for graduate students and researchers. Additionally, since the solution of many nonlinear problems can be reduced to the solution of a sequence of QP problems, it can also be used as a convenient introduction to nonlinear programming.
Variational Methods For Strongly Indefinite Problems
Author: Yanheng Ding
Publisher: World Scientific
Published: 2007-07-30
Total Pages: 177
ISBN-13: 9814474509
DOWNLOAD EBOOKThis unique book focuses on critical point theory for strongly indefinite functionals in order to deal with nonlinear variational problems in areas such as physics, mechanics and economics. With the original ingredients of Lipschitz partitions of unity of gage spaces (nonmetrizable spaces), Lipschitz normality, and sufficient conditions for the normality, as well as existence-uniqueness of flow of ODE on gage spaces, the book presents for the first time a deformation theory in locally convex topological vector spaces. It also offers satisfying variational settings for homoclinic-type solutions to Hamiltonian systems, Schrödinger equations, Dirac equations and diffusion systems, and describes recent developments in studying these problems. The concepts and methods used open up new topics worthy of in-depth exploration, and link the subject with other branches of mathematics, such as topology and geometry, providing a perspective for further studies in these areas. The analytical framework can be used to handle more infinite-dimensional Hamiltonian systems.
Lagrange Multiplier Approach to Variational Problems and Applications
Author: Kazufumi Ito
Publisher: SIAM
Published: 2008-11-06
Total Pages: 354
ISBN-13: 0898716497
DOWNLOAD EBOOKAnalyses Lagrange multiplier theory and demonstrates its impact on the development of numerical algorithms for variational problems in function spaces.
Calculus of Variations and Partial Differential Equations of First Order
Author: C. Carath‚odory
Publisher: American Mathematical Society
Published: 2024-09-30
Total Pages: 418
ISBN-13: 1470478978
DOWNLOAD EBOOKFrom the Preface: The book consists of two parts. In the first part, I have made an attempt to simplify the presentation of the theory of partial differential equations to the first order so that its study will require little time and also be accessible to the average student of mathematics ? The second part, which contains the Calculus of Variations, can also be read independently if one refers back to earlier sections in Part I ? I have never lost sight of the fact that the Calculus of Variations, as it is presented in Part II, should above all be a servant of Mechanics. Therefore, I have in particular prepared everything from the very outset for treatment in multidimensional spaces. In this second English edition of Carath‚odory's famous work, the two volumes of the first edition have been combined into one (with a combination of the two indexes into a single index). There is a deep and fundamental relationship between the differential equations that occur in the calculus of variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This basic fact forms the rationale for Carath‚odory's masterpiece.
Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models
Author: F. Giannessi
Publisher: Springer Science & Business Media
Published: 2006-04-11
Total Pages: 304
ISBN-13: 0306480263
DOWNLOAD EBOOKThe 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.
