Linear Optimization and Extensions
Problems and Solutions
(Sprache: Englisch)
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...
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Klappentext zu „Linear Optimization and Extensions “
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.
Inhaltsverzeichnis zu „Linear Optimization and Extensions “
1 Introduction1.1 Minicases and Exercises
2 The Linear Programming Problem
2.1 Exercises
3 Basic Concepts
3.1 Exercises
4 Five Preliminaries
4.1 Exercises
5 Simplex Algorithms
5.1 Exercises
6 Primal-Dual Pairs
6.1 Exercises
7 Analytical Geometry
7.1 Points, Lines, Subspaces
7.2 Polyhedra, Ideal Descriptions, Cones
7.2.1 Faces, Valid Equations, Affine Hulls
7.2.2 Facets, Minimal Complete Descriptions, Quasi-Uniqueness
7.2.3 Asymptotic Cones and Extreme Rays
7.2.4 Adjacency I, Extreme Rays of Polyhedra, Homogenization
7.3 Point Sets, Affine Transformations, Minimal Generators
7.3.1 Displaced Cones, Adjacency II, Images of Polyhedra
7.3.2 Carathéodoiy, Minkowski, Weyl
7.3.3 Minimal Generators, Canonical Generators, Quasi-Uniqueness
7.4 Double Description Algorithms
7.4.1 Correctness and Finiteness of the Algorithm
7.4.2 Geometry, Euclidean Reduction, Analysis
7.4.3 The Basis Algorithm and All-Integer Inversion
7.4.4 An All-Integer Algorithm for Double Description
7.5 Digital Sizes of Rational Polyhedra and Linear Optimization
7.5.1 Facet Complexity, Vertex Complexity, Complexity of Inversion
7.5.2 Polyhedra and Related Polytopes for Linear Optimization
7.5.3 Feasibility, Binary Search, Linear Optimization
7.5.4 Perturbation, Uniqueness, Separation
7.6 Geometry and Complexity of Simplex Algorithms
7.6.1 Pivot Column Choice, Simplex Paths, Big M Revisited
7.6.2 Gaussian Elimination, Fill-In, Scaling
7.6.3 Iterative Step I, Pivot Choice, Cholesky Factorization
7.6.4 Cross Multiplication, Iterative Step II, Integer Factorization
7.6.5 Division Free Gaussian Elimination and Cramer's Rule
7.7 Circles, Spheres, Ellipsoids
7.8 Exercises
8 Projective Algorithms
8.1 A Basic Algorithm
8.1.1 The Solution of the Approximate Problem
8.1.2 Convergence of the Approximate Iterates
8.1.3 Correctness, Finiteness, Initialization
8.2 Analysis, Algebra, Geometry
8.2.1 Solution to the Problem in the Original Space
8.2.2 The
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Solution in the Transformed Space
8.2.3 Geometric Interpretations and Properties
8.2.4 Extending the Exact Solution and Proofs
8.2.5 Examples of Projective Images
8.3 The Cross Ratio
8.4 Reflection on a Circle and Sandwiching
8.4.1 The Iterative Step
8.5 A Projective Algorithm
8.6 Centers, Barriers, Newton Steps
8.6.1 A Method of Centers
8.6.2 The Logarithmic Barrier Function
8.6.3 A Newtonian Algorithm
8.7 Exercises
9 Ellipsoid Algorithms
9.1 Matrix Norms, Approximate Inverses, Matrix Inequalities
9.2 Ellipsoid "Halving" in Approximate Arithmetic
9.3 Polynomial-Time Algorithms for Linear Programming
9.4 Deep Cuts, Sliding Objective, Large Steps, Line Search
9.4.1 Linear Programming the Ellipsoidal Way: Two Examples
9.4.2 Correctness and Finiteness of the DCS Ellipsoid Algorithm
9.5 Optimal Separators, Most Violated Separators, Separation
9.6 ?-Solidification of Flats, Polytopal Norms, Rounding
9.6.1 Rational Rounding and Continued Fractions
9.7 Optimization and Separation
9.7.1 ?-Optimal Sets and ?-Optimal Solutions
9.7.2 Finding Direction Vectors in the Asymptotic Cone
9.7.3 A CCS Ellipsoid Algorithm
9.7.4 Linear Optimization and Polyhedral Separation
9.8 Exercises
10 Combinatorial Optimization: An Introduction
10.1 The Berlin Airlift Model Revisited
10.2Complete Formulations and Their Implications
10.3 Extremal Characterizations of Ideal Formulations
10.4 Polyhedra with the Integrality Property
10.5 Exercises
- Appendices
- A Short-Term Financial Management
- A. 1 Solution to the Cash Management Case
- B Operations Management in a Refinery
- B.l Steam Production in a Refinery
- B.2 The Optimization Problem
- B.3 Technological Constraints, Profits and Costs
- B.4 Formulation of the Problem
- B.5 Solution to the Refinery Case
- C Automatized Production: PCBs and Ulysses' Problem
- C.l Solutions to Ulysses' Problem
8.2.3 Geometric Interpretations and Properties
8.2.4 Extending the Exact Solution and Proofs
8.2.5 Examples of Projective Images
8.3 The Cross Ratio
8.4 Reflection on a Circle and Sandwiching
8.4.1 The Iterative Step
8.5 A Projective Algorithm
8.6 Centers, Barriers, Newton Steps
8.6.1 A Method of Centers
8.6.2 The Logarithmic Barrier Function
8.6.3 A Newtonian Algorithm
8.7 Exercises
9 Ellipsoid Algorithms
9.1 Matrix Norms, Approximate Inverses, Matrix Inequalities
9.2 Ellipsoid "Halving" in Approximate Arithmetic
9.3 Polynomial-Time Algorithms for Linear Programming
9.4 Deep Cuts, Sliding Objective, Large Steps, Line Search
9.4.1 Linear Programming the Ellipsoidal Way: Two Examples
9.4.2 Correctness and Finiteness of the DCS Ellipsoid Algorithm
9.5 Optimal Separators, Most Violated Separators, Separation
9.6 ?-Solidification of Flats, Polytopal Norms, Rounding
9.6.1 Rational Rounding and Continued Fractions
9.7 Optimization and Separation
9.7.1 ?-Optimal Sets and ?-Optimal Solutions
9.7.2 Finding Direction Vectors in the Asymptotic Cone
9.7.3 A CCS Ellipsoid Algorithm
9.7.4 Linear Optimization and Polyhedral Separation
9.8 Exercises
10 Combinatorial Optimization: An Introduction
10.1 The Berlin Airlift Model Revisited
10.2Complete Formulations and Their Implications
10.3 Extremal Characterizations of Ideal Formulations
10.4 Polyhedra with the Integrality Property
10.5 Exercises
- Appendices
- A Short-Term Financial Management
- A. 1 Solution to the Cash Management Case
- B Operations Management in a Refinery
- B.l Steam Production in a Refinery
- B.2 The Optimization Problem
- B.3 Technological Constraints, Profits and Costs
- B.4 Formulation of the Problem
- B.5 Solution to the Refinery Case
- C Automatized Production: PCBs and Ulysses' Problem
- C.l Solutions to Ulysses' Problem
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Bibliographische Angaben
- Autoren: Dimitris Alevras , Manfred W. Padberg
- 2001, 449 Seiten, Maße: 24,2 cm, Kartoniert (TB), Englisch
- Verlag: Springer
- ISBN-10: 3540417443
- ISBN-13: 9783540417446
Sprache:
Englisch
Rezension zu „Linear Optimization and Extensions “
From the reviews: "[...] 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 präsentierten Übungsaufgaben wurden überarbeitet, ergänzt, und werden im vorliegenden Text samt Lösungen und geraffter Zusammenfassung der notwendigen theoretischen Resultate präsentiert. ... Bemerkenswert ist weiters, daß auch Programmieraufgaben gestellt und gelöst werden. ... Insgesamt ist das Buch sowohl als Quelle für Übungsaufgaben zu Vorlesungen über Lineare Optimierung als auch zum Selbststudieum sehr gut geeignet." F.Rendl (Klagenfurt), IMN - Internationale Mathematische Nachrichten 190, 2002, S.
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