Convex Analysis And Minimization Algorithms I

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Author: Jean-Baptiste Hiriart-Urruty
Publisher: Springer Science & Business Media
ISBN: 9783540568506
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Convex Analysis And Minimization Algorithms I by Jean-Baptiste Hiriart-Urruty


Original Title: Convex Analysis And Minimization Algorithms I

Convex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those adapted to non-differentiable functions, provide an immediate application of convex analysis to various fields related to optimization and operations research. These two topics making up the title of the book, reflect the two origins of the authors, who belong respectively to the academic world and to that of applications. Part I can be used as an introductory textbook (as a basis for courses, or for self-study); Part II continues this at a higher technical level and is addressed more to specialists, collecting results that so far have not appeared in books.

Inequality Problems In Mechanics And Applications

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Author: P.D. Panagiotopoulos
Publisher: Springer Science & Business Media
ISBN: 1461251524
Size: 45.46 MB
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Inequality Problems In Mechanics And Applications by P.D. Panagiotopoulos


Original Title: Inequality Problems In Mechanics And Applications

In a remarkably short time, the field of inequality problems has seen considerable development in mathematics and theoretical mechanics. Applied mechanics and the engineering sciences have also benefitted from these developments in that open problems have been treated and entirely new classes of problems have been formulated and solved. This book is an outgrowth of seven years of seminars and courses on inequality problems in mechanics for a variety of audiences in the Technical University of Aachen, the Aristotle University of Thessaloniki, the University of Hamburg and the Technical University of Milan. The book is intended for a variety of readers, mathematicians and engineers alike, as is detailed in the Guidelines for the Reader. It goes without saying that the work of G. Fichera, J. L. Lions, G. Maier, J. J. Moreau in originating and developing the theory of inequality problems has considerably influenced the present book. I also wish to acknowledge the helpful comments received from C. Bisbos, J. Haslinger, B. Kawohl, H. Matthies, H. O. May, D. Talaslidis and B. Werner. Credit is also due to G. Kyriakopoulos and T. Mandopoulou for their exceptionally diligent work in the preparation of the fmal figures. Many thanks are also due to T. Finnegan and J. Gateley for their friendly assistance from the linguistic standpoint. I would also like to thank my editors in Birkhiiuser Verlag for their cooperation, and all those who helped in the preparation of the manuscript.

Convex Analysis And Minimization Algorithms I

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Author: Jean-Baptiste Hiriart-Urruty
Publisher: Springer
ISBN: 9783540568506
Size: 14.32 MB
Format: PDF
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Convex Analysis And Minimization Algorithms I by Jean-Baptiste Hiriart-Urruty


Original Title: Convex Analysis And Minimization Algorithms I

Convex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those adapted to non-differentiable functions, provide an immediate application of convex analysis to various fields related to optimization and operations research. These two topics making up the title of the book, reflect the two origins of the authors, who belong respectively to the academic world and to that of applications. Part I can be used as an introductory textbook (as a basis for courses, or for self-study); Part II continues this at a higher technical level and is addressed more to specialists, collecting results that so far have not appeared in books.

Lagrange Type Functions In Constrained Non Convex Optimization

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Author: Alexander M. Rubinov
Publisher: Springer Science & Business Media
ISBN: 1441991727
Size: 63.54 MB
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Lagrange Type Functions In Constrained Non Convex Optimization by Alexander M. Rubinov


Original Title: Lagrange Type Functions In Constrained Non Convex Optimization

Lagrange and penalty function methods provide a powerful approach, both as a theoretical tool and a computational vehicle, for the study of constrained optimization problems. However, for a nonconvex constrained optimization problem, the classical Lagrange primal-dual method may fail to find a mini mum as a zero duality gap is not always guaranteed. A large penalty parameter is, in general, required for classical quadratic penalty functions in order that minima of penalty problems are a good approximation to those of the original constrained optimization problems. It is well-known that penaity functions with too large parameters cause an obstacle for numerical implementation. Thus the question arises how to generalize classical Lagrange and penalty functions, in order to obtain an appropriate scheme for reducing constrained optimiza tion problems to unconstrained ones that will be suitable for sufficiently broad classes of optimization problems from both the theoretical and computational viewpoints. Some approaches for such a scheme are studied in this book. One of them is as follows: an unconstrained problem is constructed, where the objective function is a convolution of the objective and constraint functions of the original problem. While a linear convolution leads to a classical Lagrange function, different kinds of nonlinear convolutions lead to interesting generalizations. We shall call functions that appear as a convolution of the objective function and the constraint functions, Lagrange-type functions.

Optimization On Low Rank Nonconvex Structures

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Author: Hiroshi Konno
Publisher: Springer Science & Business Media
ISBN: 1461540984
Size: 52.17 MB
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Optimization On Low Rank Nonconvex Structures by Hiroshi Konno


Original Title: Optimization On Low Rank Nonconvex Structures

Global optimization is one of the fastest developing fields in mathematical optimization. In fact, an increasing number of remarkably efficient deterministic algorithms have been proposed in the last ten years for solving several classes of large scale specially structured problems encountered in such areas as chemical engineering, financial engineering, location and network optimization, production and inventory control, engineering design, computational geometry, and multi-objective and multi-level optimization. These new developments motivated the authors to write a new book devoted to global optimization problems with special structures. Most of these problems, though highly nonconvex, can be characterized by the property that they reduce to convex minimization problems when some of the variables are fixed. A number of recently developed algorithms have been proved surprisingly efficient for handling typical classes of problems exhibiting such structures, namely low rank nonconvex structures. Audience: The book will serve as a fundamental reference book for all those who are interested in mathematical optimization.

Moments Positive Polynomials And Their Applications

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Author: Jean-Bernard Lasserre
Publisher: World Scientific
ISBN: 1848164467
Size: 64.48 MB
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Moments Positive Polynomials And Their Applications by Jean-Bernard Lasserre


Original Title: Moments Positive Polynomials And Their Applications

Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones, standard duality in convex optimization nicely expresses the duality between moments and positive polynomials. In the second part, the methodology is particularized and described in detail for various applications, including global optimization, probability, optimal control, mathematical finance, multivariate integration, etc., and examples are provided for each particular application. Errata(s). Errata. Sample Chapter(s). Chapter 1: The Generalized Moment Problem (227 KB). Contents: Moments and Positive Polynomials: The Generalized Moment Problem; Positive Polynomials; Moments; Algorithms for Moment Problems; Applications: Global Optimization over Polynomials; Systems of Polynomial Equations; Applications in Probability; Markov Chains Applications; Application in Mathematical Finance; Application in Control; Convex Envelope and Representation of Convex Sets; Multivariate Integration; Min-Max Problems and Nash Equilibria; Bounds on Linear PDE. Readership: Postgraduates, academics and researchers in mathematical programming, control and optimization.

Linear And Nonlinear Functional Analysis With Applications

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Author: Philippe G. Ciarlet
Publisher: SIAM
ISBN: 1611972582
Size: 71.84 MB
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Linear And Nonlinear Functional Analysis With Applications by Philippe G. Ciarlet


Original Title: Linear And Nonlinear Functional Analysis With Applications

This single-volume textbook covers the fundamentals of linear and nonlinear functional analysis, illustrating most of the basic theorems with numerous applications to linear and nonlinear partial differential equations and to selected topics from numerical analysis and optimization theory. This book has pedagogical appeal because it features self-contained and complete proofs of most of the theorems, some of which are not always easy to locate in the literature or are difficult to reconstitute. It also offers 401 problems and 52 figures, plus historical notes and many original references that provide an idea of the genesis of the important results, and it covers most of the core topics from functional analysis.

Mathematical Analysis And Applications

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Author: V. Rădulescu
Publisher: Amer Inst of Physics
ISBN: 9780735403284
Size: 47.52 MB
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Mathematical Analysis And Applications by V. Rădulescu


Original Title: Mathematical Analysis And Applications

In these proceedings from the conference in September 2005, participants describe their work at the interface between mathematical physics, numerical analysis, optimal control and calculus of variations. Papers include work in nonlinear analysis and partial differential equations as well as classical mathematical analysis and dynamical systems. Ind

Projectors And Projection Methods

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Author: Aurél Galántai
Publisher: Springer Science & Business Media
ISBN: 1441991808
Size: 13.78 MB
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Projectors And Projection Methods by Aurél Galántai


Original Title: Projectors And Projection Methods

The projectors are considered as simple but important type of matrices and operators. Their basic theory can be found in many books, among which Hal mas [177], [178] are of particular significance. The projectors or projections became an active research area in the last two decades due to ideas generated from linear algebra, statistics and various areas of algorithmic mathematics. There has also grown up a great and increasing number of projection meth ods for different purposes. The aim of this book is to give a unified survey on projectors and projection methods including the most recent results. The words projector, projection and idempotent are used as synonyms, although the word projection is more common. We assume that the reader is familiar with linear algebra and mathemati cal analysis at a bachelor level. The first chapter includes supplements from linear algebra and matrix analysis that are not incorporated in the standard courses. The second and the last chapter include the theory of projectors. Four chapters are devoted to projection methods for solving linear and non linear systems of algebraic equations and convex optimization problems.

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