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LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 89 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
The Many Facets of Linear Programming
, 2000
"... . We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction A ..."
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Cited by 25 (1 self)
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. We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium that took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the fortyfifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twentyfifth of the awarding of the 1975 Nobe...
Linear equations, Inequalities, Linear Programs (LP), and a New Efficient Algorithm
 Pages 136 in Tutorials in OR, INFORMS
, 2006
"... The dawn of mathematical modeling and algebra occurred well over 3000 years ago in several countries (Babylonia, China, India,...). The earliest algebraic systems constructed are systems of linear equations, and soon after, the famous elimination method for solving them was discovered in China and I ..."
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Cited by 7 (7 self)
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The dawn of mathematical modeling and algebra occurred well over 3000 years ago in several countries (Babylonia, China, India,...). The earliest algebraic systems constructed are systems of linear equations, and soon after, the famous elimination method for solving them was discovered in China and India. This effort culminated in the writing of two books that attracted international attention by the Arabic mathematician Muhammad ibnMusa Alkhawarizmi in the firsthalfof9thcentury. The first, AlMaqala fi Hisab aljabr w’almuqabilah (An essay on Algebra and equations), was translated into Latin under the title Ludus Algebrae, the name “algebra ” for the subject came from this Latin title, and Alkhawarizmi is regarded as the father of algebra. Linear algebra is the branch of algebra dealing with systems of linear equations. The second book KitabalJam’awalTafreeqbilHisabalHindiappeared in Latin translation under the title Algoritmi de Numero Indorum (meaning
Presolving for Semidefinite Programs Without Constraint Qualifications
, 1998
"... Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack ..."
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Cited by 2 (0 self)
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Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack of a constraint qualification which can result in both theoretical and numerical difficulties. A nonzero duality gap can exist which nullifies the elegant and powerful duality theory. Small perturbations can result in infeasibility and/or large perturbations in solutions. Such problems fall into the class of illposed problems. It is interesting to note that classes of problems where constraint qualifications fail arise from semidefinite programming relaxations of hard combinatorial problems. We look at several such classes and present two approaches to find regularized solutions. Some preliminary numerical results are included. Contents 1 Introduction 2 1.1 Notation . . . . . . . . . . ....
126 New Mathematical Disciplines and Research in the Wake of World War II
"... This paper focuses on the significance of the Second World War for the rise and establishment of new disciplines in applied mathematics as well as for the renewed interest and growth in some related subjects in pure mathematics. The mathematical topics involved are mathematical programming, operatio ..."
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This paper focuses on the significance of the Second World War for the rise and establishment of new disciplines in applied mathematics as well as for the renewed interest and growth in some related subjects in pure mathematics. The mathematical topics involved are mathematical programming, operations research, game theory, the theory of convexity, and the theory of systems of linear inequalities. Connections and interactions between different branches of mathematics on the one hand and between different kinds of driving forces in the development of mathematics on the other hand are discussed. Special emphasis is devoted to the significance of the interplay between practical problem solving and basic research in mathematics proper as a consequence of World War II and the postwar organization of science support in the USA. 1
ISBN 0 521 83378 7 hardbackFor
"... c ○ Cambridge University Press 2004 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without ..."
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c ○ Cambridge University Press 2004 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without
Library of Congress CataloguinginPublication data
"... Information on this title: www.cambridge.org/9780521833783 ..."
Typeset in Computer Modern Roman using L ATEX A catalogue record for this book is available from the British Library Library of Congress CataloguinginPublication data
"... c ○ Cambridge University Press 2004 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without ..."
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c ○ Cambridge University Press 2004 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without
Graph Theoretical Problems and Related Polytopes: Stable Sets and Perfect Graphs
"... 1.1.1 Königsberg’s bridges................... 1 ..."