Linear Optimization and Extensions

Problems and Solutions

By: Dimitris Alevras, Manfred W. Padberg

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Paperback | 11 June 2001

At a Glance

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464 Pages

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23.5 x 19.05 x 2.54

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Books on a technical topic - like linear programming - without exercises ignore the principal beneficiary of the endeavor of writing a book, namely the student - who learns best by doing course. Books with exercises - if they are challenging or at least to some extent so exercises, of - need a solutions manual so that students can have recourse to it when they need it. Here we give solutions to all exercises and case studies of M. Padberg's Linear Optimization and Exten sions (second edition, Springer-Verlag, Berlin, 1999). In addition we have included several new exercises and taken the opportunity to correct and change some of the exercises of the book. Here and in the main text of the present volume the terms "book", "text" etc. designate the second edition of Padberg's LPbook and the page and formula references refer to that edition as well. All new and changed exercises are marked by a star * in this volume. The changes that we have made in the original exercises are inconsequential for the main part of the original text where several ofthe exercises (especiallyin Chapter 9) are used on several occasions in the proof arguments. None of the exercises that are used in the estimations, etc. have been changed.

Industry Reviews

From the reviews of the first edition:

"... This book is a useful supplement to the textbook [W.Padberg, Linear Optimization and Extensions, 2nd ed., Springer, Berlin 1999]. It serves the purpose well to train MATHEMATICAL optimizers, but has little impact on the education of mathematical OPTIMIZERS, i.e. it concentrates on mathematics, but not on problem solving in reality. It is a valuable contribution for students in mathematics, but may be less suitable for students of economics or business administration."

OR-Spektrum, Issue 14, p.37, 2002

"Die in diesem Werk prasentierten Übungsaufgaben wurden uberarbeitet, erganzt, und werden im vorliegenden Text samt Losungen und geraffter Zusammenfassung der notwendigen theoretischen Resultate prasentiert. ... Bemerkenswert ist weiters, daß auch Programmieraufgaben gestellt und gelost werden. ... Insgesamt ist das Buch sowohl als Quelle fur Übungsaufgaben zu Vorlesungen uber Lineare Optimierung als auch zum Selbststudieum sehr gut geeignet."

F.Rendl (Klagenfurt), IMN - Internationale Mathematische Nachrichten 190, 2002, S. 76-77

"Do you know M. Padberg's Linear Optimization and Extensions (second edition, Springer-Verlag, Berlin, 1999)? If you teach a course on linear programming then you should know it. ... Now here is the continuation of it, discussing the solutions of all its exercises and with detailed analysis of the applications mentioned. ... For those who cherish the original textbook (students and lecturers) this is an extremely valuable sequel. For those who strive for good exercises and case studies for LP this is an excellent volume." (Peter Hajnal, Acta Scientiarum Mathematicarum, Vol.69, 2003)

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Introduction	p. 1
Minicases and Exercises	p. 1
The Linear Programming Problem	p. 39
Exercises	p. 40
Basic Concepts	p. 47
Exercises	p. 49
Five Preliminaries	p. 55
Exercises	p. 57
Simplex Algorithms	p. 63
Exercises	p. 68
Primal-Dual Pairs	p. 93
Exercises	p. 100
Analytical Geometry	p. 125
Points, Lines, Subspaces	p. 125
Polyhedra, Ideal Descriptions, Cones	p. 127
Faces, Valid Equations, Affine Hulls	p. 128
Facets, Minimal Complete Descriptions, Quasi-Uniqueness	p. 129
Asymptotic Cones and Extreme Rays	p. 130
Adjacency I, Extreme Rays of Polyhedra, Homogenization	p. 130
Point Sets, Affine Transformations, Minimal Generators	p. 131
Displaced Cones, Adjacency II, Images of Polyhedra	p. 132
Carathéodory, Minkowski, Weyl	p. 133
Minimal Generators, Canonical Generators, Quasi-Uniqueness	p. 133
Double Description Algorithms	p. 135
Correctness and Finiteness of the Algorithm	p. 136
Geometry, Euclidean Reduction, Analysis	p. 137
The Basis Algorithm and All-Integer Inversion	p. 138
An All-Integer Algorithm for Double Description	p. 139
Digital Sizes of Rational Polyhedra and Linear Optimization	p. 140
Facet Complexity, Vertex Complexity, Complexity of Inversion	p. 141
Polyhedra and Related Polytopes for Linear Optimization	p. 142
Feasibility, Binary Search, Linear Optimization	p. 142
Perturbation, Uniqueness, Separation	p. 144
Geometry and Complexity of Simplex Algorithms	p. 146
Pivot Column Choice, Simplex Paths, Big M Revisited	p. 147
Gaussian Elimination, Fill-In, Scaling	p. 148
Iterative Step I, Pivot Choice, Cholesky Factorization	p. 149
Cross Multiplication, Iterative Step II, Integer Factorization	p. 150
Division Free Gaussian Elimination and Cramer's Rule	p. 151
Circles, Spheres, Ellipsoids	p. 153
Exercises	p. 156
Projective Algorithms	p. 201
A Basic Algorithm	p. 203
The Solution of the Approximate Problem	p. 203
Convergence of the Approximate Iterates	p. 205
Correctness, Finiteness, Initialization	p. 206
Analysis,Algebra,Geometry	p. 207
Solution to the Problem in the Original Space	p. 207
The Solution in the Transformed Space	p. 209
Geometric Interpretations and Properties	p. 211
Extending the Exact Solution and Proofs	p. 214
Examples of Projective Images	p. 215
The Cross Ratio	p. 215
Reflection on a Circle and Sandwiching	p. 218
The Iterative Step	p. 220
AProjectiveAlgorithm	p. 221
Centers, Barriers, Newton Steps	p. 223
A Method of Centers	p. 224
The Logarithmic Barrier Function	p. 226
A Newtonian Algorithm	p. 228
Exercises	p. 230
Ellipsoid Algorithms	p. 263
Matrix Norms, Approximate Inverses, Matrix Inequalities	p. 265
Ellipsoid "Halving" in Approximate Arithmetic	p. 266
Polynomial-Time Algorithms for Linear Programming	p. 269
Deep Cuts, Sliding Objective, Large Steps, Line Search	p. 272
Linear Programming the Ellipsoidal Way: Two Examples	p. 274
Correctness and Finiteness of the DCS Ellipsoid Algorithm	p. 277
Optimal Separators, Most Violated Separators, Separation	p. 278
-Solidification of Flats, Polytopal Norms, Rounding	p. 280
Rational Rounding and Continued Fractions	p. 282
Optimization and Separation	p. 285
-Optimal Sets and -Optimal Solutions	p. 287
Finding Direction Vectors in the Asymptotic Cone	p. 287
A CCS Ellipsoid Algorithm	p. 288
Linear Optimization and Polyhedral Separation	p. 289
Exercises	p. 293
Combinatorial Optimization: An Introduction	p. 323
The Berlin Airlift Model Revisited	p. 323
Complete Formulations and TheirImplications	p. 327
Extremal Characterizations of Ideal Formulations	p. 331
Polyhedra with the IntegralityProperty	p. 334
Exercises	p. 336
Appendices
Short-Term Financial Management	p. 359
Solution to the Cash Management Case	p. 362
Operations Management in a Refinery	p. 371
Steam Production in a Refinery	p. 371
The Optimization Problem	p. 374
Technological Constraints, Profits and Costs	p. 378
Formulation of the Problem	p. 380
Solution to the Refinery Case	p. 381
Automatized Production: PCBs and Ulysses' Problem	p. 399
Solutions to Ulysses'Problem	p. 411
Bibliography	p. 431
Index	p. 445
Table of Contents provided by Publisher. All Rights Reserved.

Linear Optimization and Extensions

Problems and Solutions

At a Glance

Paperback

Industry Reviews

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