Linear Optimization and Extensions: Problems and Solutions (Universitext)

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. Preface ... 6 Contents ... 7 1. Introduction ... 10 1.1 Minicases and Exercises ... 10 2. The Linear Programming Problem ... 47 3. Basic Concepts ... 54 4. Five Preliminaries ... 62 5. Simplex Algorithms ... 69 6. Primal-Dual Pairs ... 99 7. Analytical Geometry ... 130 7.1 Points,Lines,Subspaces ... 130 7.2 Polyhedra.Ideal Descriptions.Cones ... 132 7.2.1 Faces,Valid Equations,Affine Hulls ... 133 7.2.2 Facets, Minimal Complete Descriptions, Quasi-Uniqueness ... 134 7.2.3 Asymptotic Cones and Extreme Rays ... 135 7.2.4 Adjacency I, Extreme Rays of Polyhedra.Homogenization ... 135 7.3 Point Sets, Affine Transformations, Minimal Generators ... 136 7.3.1 Displaced Cones, Adjacency II, Images of Polyhedra ... 137 7.3.2 Caratheodory, Minkowski, Weyl ... 138 7.3.3 Minimal Generators, Canonical Generators, Quasi-Uniqueness ... 138 7.4 Double Description Algorithms ... 140 7.4.1 Correctness and Finiteness of the Algorithm ... 141 7.4.2 Geometry, Euclidean Reduction,Analysis ... 142 7.4.3 The Basis Algorithm and All-Integer Inversion ... 143 7.4.4 An All-Integer Algorithm for Double Description ... 144 7.5 Digital Sizes of Rational Polyhedra and Linear Optimization ... 145Black,notBold,notItalic,closed,TopLeftZoom,744,1,0.0 7.5.1 Facet Complexity,Vertex Complexity,Complexity of Inversion ... 146 7.5.2 Polyhedra and Related Polytopes for Linear Optimization ... 147 7.5.3 Feasibility, Binary Search, Linear Optimization ... 147 7.5.4 Perturbation, Uniqueness, Separation ... 149 7.6 Geometry and Complexity of Simplex Algorithms ... 151 7.6.1 Pivot Column Choice, Simplex Paths, Big M Revisited ... 152 7.6.2 Gaussian Elimination,Fill-In, Scaling ... 153 7.6.3 Iterative Step I,Pivot Choice, Cholesky Factorization ... 154 7.6.4 Cross Multiplication.Iterative Step II.Integer Factorization ... 155 7.6.5 Division Free Gaussian Elimination and Cramer's Rule ... 156 7.7 Circles, Spheres, Ellipsoids ... 158 7.8 Exercises ... 161 8. Projective Algorithms ... 206 8.1 A Basic Algorithm ... 208 8.1.1 The Solution of the Approximate Problem ... 208 8.1.2 Convergence of the Approximate Iterates ... 210 8.1.3 Correctness, Finiteness, Initialization ... 211 8.2 Analysis. Algebra, Geometry ... 212 8.2.1 Solution to the Problem in the Original Space ... 212 8.2.2 The Solution in the Transformed Space ... 214 8.2.3 Geometric Interpretations and Properties ... 216 8.2.4 Extending the Exact Solution and Proofs ... 219 8.2.5 Examples of Projective Images ... 220 8.3 The Cross Ratio ... 220 8.4 Reflection on a Circle and Sandwiching ... 223 8.4.1 The Iterative Step ... 225 8.5 A Projective Algorithm ... 226 8.6 Centers, Barriers, Newton Steps ... 228 8.6.1 A Method of Centers ... 229 8.6.2 The Logarithmic Barrier Function ... 231 8.6.3 A Newtonian Algorithm ... 233 8.7 Exercises ... 235 9. Ellipsoid Algorithms ... 267 9.1 Matrix Norms, Approximate Inverses, Matrix Inequalities ... 269 9.2 Ellipsoid "Halving" in Approximate Arithmetic ... 270 9.3 Polynomial-Time Algorithms for Linear Programming ... 273 9.4 Deep Cuts, Sliding Objective, Large Steps, Line Search ... 276 9.4.1 Linear Programming the Ellipsoidal Way: Two Examples ... 278 9.4.2 Correctness and Finiteness of the DCS Ellipsoid Algorithm ... 281 9.5 Optimal Separators, Most Violated Separators, Separation ... 282 9.6 ?-Solidification of Flats, Polytopal Norms, Rounding ... 284 9.6.1 Rational Rounding and Continued Fractions ... 286 9.7 Optimization and Separation ... 289 9.7.1 ?-Optimal Sets and ?-Optimal Solutions ... 291 9.7.2 Finding Direction Vectors in the Asymptotic Cone ... 291 9.7.3 A CCS Ellipsoid Algorithm ... 292 9.7.4 Linear Optimization and Polyhedral Separation ... 293 9.8 Exercises ... 297 10. Combinatorial Optimization: An Introduction ... 326 10.1 The Berlin Airlift Model Revisited ... 326 10.2 Complete Formulations and Their Implications ... 330 10.3 Extremal Characterizations of Ideal Formulations ... 334 10.4 Polyhedra with the Integrality Property ... 337 10.5 Exercises ... 339 A. Short-Term Financial Management ... 362 A.1 Solution to the Cash Management Case ... 365 B. Operations Management in a Refinery ... 373 B.1 Steam Production in a Refinery ... 373 B.2 The Optimization Problem ... 376 B.3 Technological Constraints, Profits and Costs ... 380 B.4 Formulation of the Problem ... 382 B.5 Solution to the Refinery Case ... 383 C. Automatized Production: PCBs and Ulysses' Problem ... 400 C.1 Solutions to Ulysses' Problem. ... 412 Bibliography ... 432 Index ... 445 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. Here 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. (i) To formulate LLI's problem denote by x1 the liters of bourbon and by x2 the liters of whiskey produced during the planning period.