Holdings: Linear optimization and extensions

Linear optimization and extensions: problems and solutions

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Bibliographic Details
Main Authors:	Alevras, Dimitris (Author), Padberg, Manfred W. (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2001
Series:	Universitext
Subjects:	Lineare Optimierung - Lehrbuch Linear programming > Problems, exercises, etc Mathematical optimization > Problems, exercises, etc Lineare Optimierung
Online Access:	Inhaltsverzeichnis
Physical Description:	IX, 449 S. graph. Darst.
ISBN:	3540417443

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adam_text	CONTENTS 1 INTRODUCTION 1 1.1 MINICASES AND EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 THE LINEAR PROGRAMMING PROBLEM 39 2.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 BASIC CONCEPTS 47 3.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 FIVE PRELIMINARIES 55 4.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 SIMPLEX ALGORITHMS 63 5.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6 PRIMAL-DUAL PAIRS 93 6.1 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7 ANALYTICAL GEOMETRY 125 7.1 POINTS, LINES, SUBSPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2 POLYHEDRA, IDEAL DESCRIPTIONS, CONES . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2.1 FACES, VALID EQUATIONS, AFFINE HULLS . . . . . . . . . . . . . . . . . . . . . . . . 128 7.2.2 FACETS, MINIMAL COMPLETE DESCRIPTIONS, QUASI-UNIQUENESS . . . . . . . . . . 129 7.2.3 ASYMPTOTIC CONES AND EXTREME RAYS . . . . . . . . . . . . . . . . . . . . . . . 130 7.2.4 ADJACENCY I, EXTREME RAYS OF POLYHEDRA, HOMOGENIZATION . . . . . . . . . . . 130 7.3 POINT SETS, AFFINE TRANSFORMATIONS, MINIMAL GENERATORS . . . . . . . . . . . . . . . . 131 7.3.1 DISPLACED CONES, ADJACENCY II, IMAGES OF POLYHEDRA . . . . . . . . . . . . . . 132 7.3.2 CARATH ´ EODORY, MINKOWSKI, WEYL . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.3.3 MINIMAL GENERATORS, CANONICAL GENERATORS, QUASI-UNIQUENESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.4 DOUBLE DESCRIPTION ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.4.1 CORRECTNESS AND FINITENESS OF THE ALGORITHM . . . . . . . . . . . . . . . . . . . 136 7.4.2 GEOMETRY, EUCLIDEAN REDUCTION, ANALYSIS . . . . . . . . . . . . . . . . . . . . 137 7.4.3 THE BASIS ALGORITHM AND ALL-INTEGER INVERSION . . . . . . . . . . . . . . . . . . 138 7.4.4 AN ALL-INTEGER ALGORITHM FOR DOUBLE DESCRIPTION . . . . . . . . . . . . . . . . . 139 7.5 DIGITAL SIZES OF RATIONAL POLYHEDRA AND LINEAR OPTIMIZATION . . . . . . . . . . . . . . 140 7.5.1 FACET COMPLEXITY, VERTEX COMPLEXITY, COMPLEXITY OF INVERSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.5.2 POLYHEDRA AND RELATED POLYTOPES FOR LINEAR OPTIMIZATION . . . . . . . . . . . 142 7.5.3 FEASIBILITY, BINARY SEARCH, LINEAR OPTIMIZATION . . . . . . . . . . . . . . . . . 142 7.5.4 PERTURBATION, UNIQUENESS, SEPARATION . . . . . . . . . . . . . . . . . . . . . . 144 7.6 GEOMETRY AND COMPLEXITY OF SIMPLEX ALGORITHMS . . . . . . . . . . . . . . . . . . . . 146 7.6.1 PIVOT COLUMN CHOICE, SIMPLEX PATHS, BIG M REVISITED . . . . . . . . . . . . . 147 7.6.2 GAUSSIAN ELIMINATION, FILL-IN, SCALING . . . . . . . . . . . . . . . . . . . . . . 148 VIII CONTENTS 7.6.3 ITERATIVE STEP I, PIVOT CHOICE, CHOLESKY FACTORIZATION . . . . . . . . . . . . . . 149 7.6.4 CROSS MULTIPLICATION, ITERATIVE STEP II, INTEGER FACTORIZATION . . . . . . . . . . 150 7.6.5 DIVISION FREE GAUSSIAN ELIMINATION AND CRAMERS RULE . . . . . . . . . . . . 151 7.7 CIRCLES, SPHERES, ELLIPSOIDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.8 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8 PROJECTIVE ALGORITHMS 201 8.1 A BASIC ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.1.1 THE SOLUTION OF THE APPROXIMATE PROBLEM . . . . . . . . . . . . . . . . . . . . 203 8.1.2 CONVERGENCE OF THE APPROXIMATE ITERATES . . . . . . . . . . . . . . . . . . . . . 205 8.1.3 CORRECTNESS, FINITENESS, INITIALIZATION . . . . . . . . . . . . . . . . . . . . . . . 206 8.2 ANALYSIS, ALGEBRA, GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.2.1 SOLUTION TO THE PROBLEM IN THE ORIGINAL SPACE . . . . . . . . . . . . . . . . . . 207 8.2.2 THE SOLUTION IN THE TRANSFORMED SPACE . . . . . . . . . . . . . . . . . . . . . . 209 8.2.3 GEOMETRIC INTERPRETATIONS AND PROPERTIES . . . . . . . . . . . . . . . . . . . . . 211 8.2.4 EXTENDING THE EXACT SOLUTION AND PROOFS . . . . . . . . . . . . . . . . . . . . . 214 8.2.5 EXAMPLES OF PROJECTIVE IMAGES . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.3 THE CROSS RATIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.4 REFLECTION ON A CIRCLE AND SANDWICHING . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.4.1 THE ITERATIVE STEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.5 A PROJECTIVE ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.6 CENTERS, BARRIERS, NEWTON STEPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.6.1 A METHOD OF CENTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.6.2 THE LOGARITHMIC BARRIER FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.6.3 A NEWTONIAN ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.7 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9 ELLIPSOID ALGORITHMS 263 9.1 MATRIX NORMS, APPROXIMATE INVERSES, MATRIX INEQUALITIES . . . . . . . . . . . . . . . 265 9.2 ELLIPSOID HALVING* IN APPROXIMATE ARITHMETIC . . . . . . . . . . . . . . . . . . . . . . 266 9.3 POLYNOMIAL-TIME ALGORITHMS FOR LINEAR PROGRAMMING . . . . . . . . . . . . . . . . . 269 9.4 DEEP CUTS, SLIDING OBJECTIVE, LARGE STEPS, LINE SEARCH . . . . . . . . . . . . . . . . 272 9.4.1 LINEAR PROGRAMMING THE ELLIPSOIDAL WAY: TWO EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 9.4.2 CORRECTNESS AND FINITENESS OF THE DCS ELLIPSOID ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.5 OPTIMAL SEPARATORS, MOST VIOLATED SEPARATORS, SEPARATION . . . . . . . . . . . . . . 278 9.6 * -SOLIDIFICATION OF FLATS, POLYTOPAL NORMS, ROUNDING . . . . . . . . . . . . . . . . . . 280 9.6.1 RATIONAL ROUNDING AND CONTINUED FRACTIONS . . . . . . . . . . . . . . . . . . . 282 9.7 OPTIMIZATION AND SEPARATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 9.7.1 * -OPTIMAL SETS AND * -OPTIMAL SOLUTIONS . . . . . . . . . . . . . . . . . . . . . 287 9.7.2 FINDING DIRECTION VECTORS IN THE ASYMPTOTIC CONE . . . . . . . . . . . . . . . . 287 9.7.3 A CCS ELLIPSOID ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 9.7.4 LINEAR OPTIMIZATION AND POLYHEDRAL SEPARATION . . . . . . . . . . . . . . . . . 289 9.8 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 CONTENTS IX 10 COMBINATORIAL OPTIMIZATION: AN INTRODUCTION 323 10.1THE BERLIN AIRLIFT MODEL REVISITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 10.2COMPLETE FORMULATIONS AND THEIR IMPLICATIONS . . . . . . . . . . . . . . . . . . . . . 327 10.3EXTREMAL CHARACTERIZATIONS OF IDEAL FORMULATIONS . . . . . . . . . . . . . . . . . . . 331 10.4POLYHEDRA WITH THE INTEGRALITY PROPERTY . . . . . . . . . . . . . . . . . . . . . . . . . . 334 10.5EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 APPENDICES A SHORT-TERM FINANCIAL MANAGEMENT 359 A.1 SOLUTION TO THE CASH MANAGEMENT CASE . . . . . . . . . . . . . . . . . . . . . . . . . 362 B OPERATIONS MANAGEMENT IN A REFINERY 371 B.1 STEAM PRODUCTION IN A REFINERY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 B.2 THE OPTIMIZATION PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 B.3 TECHNOLOGICAL CONSTRAINTS, PROFITS AND COSTS . . . . . . . . . . . . . . . . . . . . . . 378 B.4 FORMULATION OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 B.5 SOLUTION TO THE REFINERY CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 C AUTOMATIZED PRODUCTION: PCBS AND ULYSSES* PROBLEM 399 C.1 SOLUTIONS TO ULYSSES* PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 BIBLIOGRAPHY 431 INDEX 445
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spelling	Alevras, Dimitris Verfasser aut Linear optimization and extensions problems and solutions Dimitris Alevras ; Manfred W. Padberg Berlin [u.a.] Springer 2001 IX, 449 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Lineare Optimierung - Lehrbuch Linear programming Problems, exercises, etc Mathematical optimization Problems, exercises, etc Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 s DE-604 Padberg, Manfred W. Verfasser aut SWBplus Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009328643&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Alevras, Dimitris Padberg, Manfred W. Linear optimization and extensions problems and solutions Lineare Optimierung - Lehrbuch Linear programming Problems, exercises, etc Mathematical optimization Problems, exercises, etc Lineare Optimierung (DE-588)4035816-1 gnd
subject_GND	(DE-588)4035816-1
title	Linear optimization and extensions problems and solutions
title_auth	Linear optimization and extensions problems and solutions
title_exact_search	Linear optimization and extensions problems and solutions
title_full	Linear optimization and extensions problems and solutions Dimitris Alevras ; Manfred W. Padberg
title_fullStr	Linear optimization and extensions problems and solutions Dimitris Alevras ; Manfred W. Padberg
title_full_unstemmed	Linear optimization and extensions problems and solutions Dimitris Alevras ; Manfred W. Padberg
title_short	Linear optimization and extensions
title_sort	linear optimization and extensions problems and solutions
title_sub	problems and solutions
topic	Lineare Optimierung - Lehrbuch Linear programming Problems, exercises, etc Mathematical optimization Problems, exercises, etc Lineare Optimierung (DE-588)4035816-1 gnd
topic_facet	Lineare Optimierung - Lehrbuch Linear programming Problems, exercises, etc Mathematical optimization Problems, exercises, etc Lineare Optimierung
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