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.

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.- 7.2.1 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 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.