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.

Exercises-oriented textbook in a huge field, where elementary textbooks are rare Includes supplementary material: sn.pub/extras

Klappentext

This book offers a comprehensive treatment of the exercises and case studies as well as summaries of the chapters of the book "Linear Optimization and Extensions" by Manfred Padberg. It covers the areas of linear programming and the optimization of linear functions over polyhedra in finite dimensional Euclidean vector spaces.BRHere are the main topics treated in the book: Simplex algorithms and their derivatives including the duality theory of linear programming. Polyhedral theory, pointwise and linear descriptions of polyhedra, double description algorithms, Gaussian elimination with and without division, the complexity of simplex steps. Projective algorithms, the geometry of projective algorithms, Newtonian barrier methods. Ellipsoids algorithms in perfect and in finite precision arithmetic, the equivalence of linear optimization and polyhedral separation. The foundations of mixed-integer programming and combinatorial optimization.

Inhalt
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.- 7.3 Point Sets, Affine Transformations, Minimal Generators.- 7.4 Double Description Algorithms.- 7.5 Digital Sizes of Rational Polyhedra and Linear Optimization.- 7.6 Geometry and Complexity of Simplex Algorithms.- 7.7 Circles, Spheres, Ellipsoids.- 7.8 Exercises.- 8 Projective Algorithms.- 8.1 A Basic Algorithm.- 8.2 Analysis, Algebra, Geometry.- 8.3 The Cross Ratio.- 8.4 Reflection on a Circle and Sandwiching.- 8.5 A Projective Algorithm.- 8.6 Centers, Barriers, Newton Steps.- 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.- 9.5 Optimal Separators, Most Violated Separators, Separation.- 9.6 ?-Solidification of Flats, Polytopal Norms, Rounding.- 9.7 Optimization and 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.4Formulation of the Problem.- B.5 Solution to the Refinery Case.- C Automatized Production: PCBs and Ulysses' Problem.- C.l Solutions to Ulysses' Problem.