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Linear Optimization and Extensions


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Table of Contents

1 Introduction.- 1.1 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 Caratheodoiy, 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 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.

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Springer Book Archives


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 UEbungsaufgaben wurden uberarbeitet, erganzt, und werden im vorliegenden Text samt Loesungen und geraffter Zusammenfassung der notwendigen theoretischen Resultate prasentiert. ... Bemerkenswert ist weiters, dass auch Programmieraufgaben gestellt und geloest werden. ... Insgesamt ist das Buch sowohl als Quelle fur UEbungsaufgaben 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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