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Matrices and Simplex Algorithms CuuDuongThanCong.com A R G Heestennan Department of Economics, University of Birmingham , U.K Matrices and Simplex Algorithms A Textbook in Mathematical Programming and Its Associated Mathematical Topics D Reidel Publishing Company Dordrecht : Holland / Boston : U.S.A / London: England CuuDuongThanCong.com library of Congress Cataloging in Publication Data Heesterman, A R G Matrices and simplex algorithms Includes index Programming (Mathematics) QA402.5.H43 1982 519.7 ISBN-13:978-94-009-7943-7 DOl: 10.1 007/978-94-009-7941-3 Ma trices I 82-18540 Title e-ISBN -13: 978-94-009-7941-3 Published by D Reidel Publishing Company P.O Box 17,3300 AA Dordrecht, Holland Sold and distributed in the U.S.A and Canada by Kluwer Boston Inc., 190 Old Derby Street, Hingham, MA 02043, U.S.A In all other countries, sold and distribu ted by Kluwer Academic Publishers Group, P.O Box 322,3300 AH Dordrecht, Holland D Reidel Publishing Company is a member of the Kluwer Group All Rights Reserved Copyright (91983 by D Reidel Publishing Company, Dordrecht, Holland Softcover reprint ofthe hardcover 1st edition 1983 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recordipg or by any informational storage and retrieval system, without written permission from the copyright owner CuuDuongThanCong.com Table of Contents Introduction vii PART I MATRICES, BLOCK-EQUATIONS, AND DETERMINANTS Chapter I / EQUATIONS-SYSTEMS AND TABLEAUX Chapter II/MATRIX NOTATION Chapter III/BLOCK-EQUATIONS AND MATRIX-INVERSION 33 Chapter IV / SOME OPERATORS AND THEIR USE 62 Chapter V / DETERMINANTS AND RANK 68 PART II GRAPHS AND LINEAR PROGRAMMING Chapter VI/VECTORS AND COORDINATE-SPACES 115 Chapter VII/SOME BASIC LINEAR PROGRAMMING CONCEPTS 144 Chapter VIII/OUTLINE OF THE SIMPLEX ALGORITHM 149 Chapter IX / THE SEARCH FOR A FEASIBLE SOLUTION 181 Chapter X / MIXED SYSTEMS, UPPER AND LOWER BOUNDS 205 Chapter XI/DUALITY 223 Chapter XII/LINEAR PROGRAMMING ON THE COMPUTER 242 Chapter XIII/PARAMETRIC VARIATION OF THE L.P PROBLEM 273 N.B Answers to exercises are not included in this list of contents, but may be found in the places indicated with the exercises, usually at the end of the chapters CuuDuongThanCong.com vi T ABLE OF CONTENTS PART III SOME GENERAL MATHEMATICAL PROGRAMMING NOTIONS AND RELATED MATRIX ALGEBRA Chapter XIV / TOPOLOGY OF FEASIBLE SPACE AREAS AND ITS RELATION TO DEFINITENESS 319 Chapter XV / OPTIMALITY CONDITIONS 363 PART IV gUADRATIC PROGRAMMING Chapter XVI/QUADRATIC PROGRAMMING WITH LINEAR RESTRICTIONS 402 Chapter XVII/PARAMETRIC METHODS IN QUADRATIC PROGRAMMING 516 Chapter XVIII/GENERAL QUADRATIC PROGRAMMING 556 PART V INTEGER PROGRAMMING Chapter XIX / INTEGER PROGRAMMING AND SOME OF ITS APPLICATIONS 637 Chapter XX / BRANCHING METHODS 656 Chapter XXI/THE USE OF CUTS 702 REFERENCES 773 INDEX 779 CuuDuongThanCong.com Introduction This is a textbook algorithms and the algorithms It was mathematics itself devoted to mathematical programming mathematics needed to understand such mainly written for economists, but the obviously has relevance for other disciplines It is a textbook as well a~ in parts, a contribution to new knowledge There is, accordingly, a broad ordering of climbing sophistication, the earlier chapters being purely for the student, the later chapters being more specialist and containing some element of novelty on certain points The book is edited in five parts Part I deals with elementary matrix operations, matrix inversion, determinants, etc Part II is mainly devoted to linear programming As far as students' readability is concerned, these two parts are elementary undergraduate material However, I would claim, in particular with respect to linear programming, that I things more efficiently than the standard textbook approach has it This refers mainly to the search for a feasible solution i.e Chapter 9, and to upper and lower limits, i.e Chapter 10 I have also argued that the standard textbook treatment of degeneracy misses a relevant problem, namely that of accuracy In short, I would invite anyone who has the task of writing or designing an LP-code, to first acquaint himself with my ideas CuuDuongThanCong.com viii INTRODUCTION Parts III and IV are concerned with nonlinear programming Part III gives the bulk of the theory in general terms including additional matrix algebra It was obviously necessary to introduce definiteness at this point, but a full discussion of latent roots is refrained from Proofs are therefore given as far as possible, without reference to eigenvalues However, certain results will have to be taken on trust by those readers who have no prior knowledge on this point The main contribution to the literature made in part III, probably is Chapter 15, i.e to explain both the first-order conditions and the second order conditions for a constrained maximum, in terms where one may expect the student to actually understand this admittedly difficult problem The unconventional concept of subspace convexity is not, and cannot be a true novelty; it is equivalent to the more usual way of formulating the second order condition for a constrained maximum in terms of determinants Part IV is concerned with quadratic programming It does not give a comprehensive survey of algorithms It gives those algorithms which I considered the most efficient, and the easiest to explain and to be understood With respect to novelty, Chapters 16 and 17 not contain any original ideas or novel approaches, but some of the ideas developed in Chapters and 10 for the LP case are carried over into quadratic programming Chapter 19 does however, offer an algorithm developed by myself, concerning quadratic programming with quadratic side-conditions Part V deals with integer programming As in the QP case, the basic ingredients are taken from the existing literature, but the branching algorithm of section 20.2, although based on a well-established approach, was developed by myself Also, the use of upper and lower limits on the lines of Chapter 10 proved particularly useful in the integer programming context Nothing in this book is out of the reach of undergraduate students, but if it is to be read in its entirety by people without prior k~owledge beyond "0" level mathematics, the consecutive ordering of the material becomes essential and a two-year period of assimilation with a break between Part II and Part III would be preferable However Parts IV and V will generally be considered to be too specialist on grounds of relevance and curriculum load, and accordingly be considered more suitable for postgraduate students specializing in O.R or mathematical programming CuuDuongThanCong.com INTRODUCTION I have, however, myself used some sections of Chapter 16, not so much because undergraduates need to know quadratic programming for its own sake, but as reinforcement of teaching optimality conditions Concerning presentation, it may be observed that more than half of the number of pages is taken up by numerical examples, graphical illustrations and text-listings of programme-code The numerical examples and graphical illustrations are obviously there for purely educational purposes as are the student exercises The code-listings serve, however a dual purpose to back up the descriptions of algorithms, and to be available for computational use Some tension between these two purposes is obviously unavoidable I have made an effort to make the programme-texts readable, not only for the machine but also for the human reader, but if illustration for the benefit of the human reader were the only consideration I would have to cut down on the number of pages of code-listing much more than I have in fact done It appears to be appropriate to comment here on the use of the computer-language i.e Algol 60, rather than the more widely used Fortran While it is true that I simply know Algol very much better than Fortran, the choice appears also to be justified on the following intrinsic grounds: The use of alphanumerical labels which are meaningful to the human reader, e.g PHASE I:, MAKE THE STEP:, etc., and corresponding goto statements e.g 'COTO' PHASE I; helps to bridge the gap between programme description and programme-text in a way which is difficult to achieve by comment (or its Fortran equivalent) only Many procedure and programme texts also contain alphanumerical labels which are there purely for the human reader, as there are no corresponding goto statements While such labels are also possible in an "Algol-like" language as, for example Pascal, they cannot be used in Fortran Furthermore, Fortran is simply more primitive than Algol CuuDuongThanCong.com ix INTRODUCTION x I have made extensive use of block-structure and dynamic arrays and as a result my programmes don:t require more core-space than is strictly necessary Also, I gather that not all versions of Fortran permit recursive calls in the way I have used them in section 5.6 for the calculation of determinants and in section 20.4 for branching in integer programming The value of the text-listings as a direct source of ready-made programme-text is further compromised by the presence of warning-messages, not only in the main programmes but also inside the procedures The presence of these warning messages enhances the readability for the human reader but, as they are system-specific, they will in general require adaptation if the algorithms are to be applied in a different machine-environment Acknowledgement I gladly acknowledge the use of the facilities of the University of Birmingham Computer Center, as well as the help of Dr P Soldutos, my student at the time, in setting up the private graph-plotting package used to make the graphs in this book University of Birmingham 2ls t May, 1982 A.R.C Heesterman CuuDuongThanCong.com Part I MATRICES, BLOCK-EQUATIONS, AND DETERMINANTS CHAPTER I EQUATIONS-SYSTEMS AND TABLEAUX 1.1 1.2 Equations-systems The use of a tableau 3 CHAPTER II MATRIX NOTATION 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 The purpose of matrix notation Some definitions and conventions The transpose of a matrix Addition and subtraction Matrix multiplication The product of a matrix and a vector Vector vector multiplication Multiplication by a scalar Matrix-matrix mUltiplication by columns or by rows Substitution Forms Recursive products The addition of several matrices Transposition of compound expressions Some special mat~ices and vectors Matrix partitioning Multiplication by partitionins Differentiation of matrix expressions Reading and printing of large matrices by electronic computers CuuDuongThanCong.com 5 10 II 13 14 14 15 16 17 18 19 19 21 22 25 29 REFERENCES 776 [23] Kim, C Introduction to Linear Programming New York, Rinehart and Winston, 1971 Referred to: 11.3 [24] Kuhn, R.W., and Tucker, A.W Nonlinear Programming Tn: Se.cond Berkeley SY.lllPosium on Mathematics and Probability Theory Univers.ity of California Press, Berkeley, Calif., U S A , 1951, pp 481-492 Also reprinted in: Ne~an, P (Editor) Reading& in Mathematical Economics The Johns Hopkins Press, Baltimore, U.S.A., 1968 Vol 1, pp.3-l4 Referred to: 15.3 [25] La Grange (de la Grange) Mechanique Analytique (Section Quatri~e) Paris, Veuve Desaint, 1788 Reprinted in: Oeuvres de la Grange Paris, Gauthier et fils, 1888 (Tome XL) Referred to: 15.3 [26] Land, A.R., and Doig, A.G An Automatic Method for Solving Discrete Programming Problems Econometrica 28, pp.497-520, 1960 Referred to: :20.2, 20.3 ° [27] Mangasarin, L Nonlinear Programming New York, McGraw Hill, 1969 Referred to: 14.2 [28] Maurer, S Pivotal Theory of Determinants In: Balinski, M.L Pivoting and Extensions: In Honour of A.W Tucker Amsterdam, North Holland Publishing Co., 1974 Referred to: 5.5 [29] Parlett, B.N The Symmetric Eigenvalue Problem Prentice Hall, Englewood Cliffs, New Jersey, U.S.A., 1980 Referred to: 14.4 CuuDuongThanCong.com REFERENCES [30] Ponstein, J Seven Types of Convexity Siam Review g (1967), pp.115-ll9 Referred to:- 14.2 [31] Powell, M.J.D A Fast Algorithm for Nonlinearly Constrained Optimization Calculations In: Numerical Analysis Conference, Dundee, 1977, Siam Springer Verlag Referre.d to: 18.2 [32] Samuels.on, P.A of Economic Analysis Harvard University Press, Cambridge, Mass., U.S.A., 1967 Referred to: 15.5 Foundation~ [ 33 J Theil, H Principles of Econometrics New York, John Wiley & Amsterdam, N.Holland Pubg Co., 1971 Referre.d to: 14.4 [34] Tucker, A.W Dual Sys.tems of Homogeneous Linear Inequalities In: Kuhn, H.W , and Tucker, A.W (Editors} Linear Inequalities and Related Systems Princeton Uniyersity Press, New Jersey, U.S.A., 1965 Referred to: 11.3 [35] Van de Panne, C Methods of Linear and Quadratic Programming Amsterdam, North Holland Publi.s.hing Co., 19]5 Re.ferred to: 8.10,17.6, lJ.6 [36] Van de Panne, C., and Whinston, A The Simplex and Dual Met.hod for Quadratic Programming Operational Research Quarterly, !2, Nr 4, pp 355-38] Referred to: 16.2, 16.3 [37] Van de Panne, C., and Whinston, A Simplicial Methods in Quadratic Programming Naval Res.earch Logistics Quarterly, (1964) pp 2]3-302 Referred to: 16.3 CuuDuongThanCong.com 777 77 REFERENCES [38] Weingartner, M Mathematical Programming and the Analysis of Capital Budgeting Problems Prentice Hall, Englewood Cliffs, New Jersey, U.S.A., 1963 Referred to: 20.1 [39] Westphal, L.E Planning Investment Decisions with Economies of Scale North Holland Publishing Co., 1971 Referred to: 19.2 [40] Whinston, A Some Implications of the Conjugate Function Theory to Duality In Abadie [1], 1967, pp.77-96 Referred to: 15.6 [41] Zionts, S Linear and Integer Programming Prentice Hall, Englewood Cliffs, New Jersey, U.S.A., 1974 Referred to: 21.1 [42] Zoutendijk, G Mathematical Programming Methods Amsterdam, North Holland Publishing Co., 1975 Referred to: 8.10 CuuDuongThanCong.com I N D E X Accuracy 166 addition of matrices of several matrices 18 additive property of restrictions 363 adjoint, all-integer elimination and 101 aggregate restriction 370, 373, 586 aggregation matrix 63 ALGOL 60 ix all integer programming problem 637 amended convex mode operation 473, 480 ample fulfillment of restriction 148 anticonvex function, constrained maximum of 358 restriction 320, 587 approximation, initial, inadequate 562 loose 576 overtight 564 tangential 583, 590 artificial right-hand side 546 variables 181, 199 Ay = Bx system 40 Badname restriction 197, 238 row 197 variable 412 basic variable 150 basis matrix 151, 161 inverse of 162 basis variables, free 249 binding restriction, linear 570 in Lagrangeans 378 779 CuuDuongThanCong.com 780 INDEX parametric variation and 567 block-colwnn 21 blocked incoming variable see bounded incoming variable block elimination 53 equations 33 inversion 54 pivot 151 pivoting 54 with inequalities 223 block-row 21 boundary point 140, 142 bounded incoming variable 458 boundedness and artificial feasibility 458 branch, definition 656 end 673 higher 668 lower 668 main, higher and lower 669 branching algorithm commentary 678-689 method, Dakin's 689 Land and Doig's 690 methods 656 mixed-integer programming and 668 procedure, recursive 690 restriction 668 Canonical form 149, 151 characteristic equation 341 characteristic vector 341 circle 134 code (in branching problem) 661 lowest 661 recorded 661 wipe out of 661 cofactors 69 column of matrix combination of rows 33 of vectors 124 combined restrictions, tangential equivalent of 365 complementarity rule 414 complementary slackness condition 369, 370 complex roots in quadratic programming 568, 573 composite matrix 22 composite vector 22 computational requirement, product-form inverse, revised simplex algorithm 272 computer handling of large matrices 29 computer efficiency, degeneracy and 168 conservative smallest quotient rule 191 consistency and optimality 588 CuuDuongThanCong.com INDEX constrained maximum of anticonvex function 358 second-order conditions 404 constrained problem 586 constraint qualification condition 369 continuous optimum 638 continuous problem, feasible solution 638 continuous variable 637 convergency 166 convexity of Lagrangean 388 of parametric variation 287 of quadratic function 339 convex complication 125 mode operation 473 primal boundedness 470 programming problems 320 restriction 320 set 124, 125 transition 437 coordinate plane 45 spaces 115 system, secondary 137 correcting dual adjustment 602 correction equation 592 correction restriction 601 corresponding continuous problem (in integer programming) 638 costs, fixed 638 Cottle's algorithm 482 cubic function 133 cubic integer programming 645 cuts 702 augmented 702, 709 classes- of 714 combined 702, 711 elementary 702, 706 integer-value 704 limit 706 lower limit 707 main 717 on variables 705 pre liminary 71 priority rules 714 subsidiary 717 upper limit cutting algorithm summary 741 Decision variables 638 decomposition of a determinant 82 definite matrix and diagonal pivoting 352 CuuDuongThanCong.com 781 782 INDEX subspace and partial inversion of 350 definiteness and topology of feasible spaces 319 degeneracy 164 dual 240 problems of 166 resolution of 169 determinantal equation (see also characteristic equation) 341 determinants 68 calculation of 87 decomposition of 82 of structured matrices 91 permutation of 75 product of two square matrices 108 diagonal matrix 20 diagonal pivoting and definite matrices 352 diagonal vector 19 differentiation of inverses 58 of matrix expressions 25 directional convexity 323, 325 displaced and constrained problem 586 displaced problem 586 distinguished variable see badname variable 412 division, multiplication instead of 48 domain 324 downward adjustment 602 driving variable 413 parameter theorem 449 parametric equivalence method 544 dual degeneracy 240 dual feasibility, computational advantage 236 duality 223 nonlinear 397 theorem 228 application of 233 dual parametric step 291 dual problem 230 ratio 237 requirement condition 370 simplex method 234 variables see also shadow prices 368, 377 as indicators of change in object function 379 as leaving variable 458 elimination of 447 in parametric variation 532 economic interpretation 380 upward adjustment of 591 dummy variable 641 Efficient row selection rule 251 eigenvalue of a matrix 341 CuuDuongThanCong.com INDEX 783 element of a matrix elimination 34 all-integer adjoint and 101 all-integer method 48 computerized procedure 37 emptiness 547 in nonconvex problems 575 empty problem 184, 659 equation without nonnegativity restriction 486 equations 207, 238 simultaneous linear examples 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 27, 34, 35, 37, 38, 40, 43, 44, 47, 48, 49, 51, 55, 56, 59, 62, 66, 68, 73, 82, 83, 85, 94, 95, 96, 103, 111, 117, 118, 120, 123, 127, 130, 134, 138, 145, 154, 165, 177, 179, 182, 188, 189, 202, 205, 208, 209, 210, 217, 218, 231, 232, 235, 240, 268, 269, 270, 271, 276, 287, 290, 293, 295, 304, 319, 321, 322, 332, 335, 340, 343, 344, 349, 353, 354, 370, 374, 375, 379, 383, 386, 392, 395, 404, 408, 411, 429, 439, 443, 4S8, 459, 462, 468, 476, 480, 487, 489, 492, 495, 526, 531, 533, 542, 546, 556, 561, 563, 567, 568, 574, 576, 583, 591, 594, 595, 599, 600, 612, 637, 639, 640, 641, 669, 703, 711, 713, 739 exercises 11, 13, 18, 25, 29, 41, 47, 59, 67, 75, 82, 86, 87, 99, 100(2),105,111,116,139,159,194,204,217,240,289, 334,355(2),382,396,415(2),419,432,447,455,473,525, 530, 613 (answer 633) extrema of quadratic function 339 extreme point 140 Factorization of semidefinite matrix 356 feasible area 148 corners of 150 feasible solution 144, 153 search for 181, 250 column selection in 237 free variable entering 249 feasible space area end bounding by 535 topology of 319 feasibility, artificial, boundedness and 458 fixed costs 638 Fortran, limitations of, in L.P 243 free variable (= unconstrained variable) 145 full exhaustion 659 fully explored problem 673 subbranch 673 function 116 argument of 116 linear, graph of 117 CuuDuongThanCong.com 784 value 116 further out subbranch problem 673 Half space 128 head problem 673 highest step rule 163, 251 hyperbola 136 hypothetical step 717 Indefinite matrix 341 roots 344 indifferent restriction 141 inequalities,' block pivoting with 223 infeasible starting solution 473 ill-conditioned systems 34 immediate neighbour convexity transmission 439 improper vertex 537 imputed prices ~ dual variables index, increasing order of, branching in 656 integer optimum 638 integer programming V111, 636 applications 637 terminology 637 integer requirement 638 integer-restricted variable 637 integer solution 638 interior point 139 inverse matrix 41 definition 43 differentiation of 58 inversion and reduction 47 by row operators 84 of matrices 33 of recursive products 56 of transposes 56 partial, of definite matrices 350 row permutation during 51 investment cos·ts 640 inward adjustment 584, 593 Kuhn-Tucker theorem 368 Lagrangeans 368 convexity of, in coordinate space 388 latent root of a matrix 341 leading variable 671 linear approximation 559 of cubic function 642 relaxation of 568 linear programming vii concepts 144 CuuDuongThanCong.com INDEX INDEX described procedure for 249 graphical solution 146 matrix notation 145 on computer 242 parametric variation 273 program text 264 linear restrictions, quadratic restrictions with 402 linear subspaces 118 linear transform of coordinates 137 lower bounds 205, 217 in quadratic programming 495 Mathematical programming, general 318 problem 319 matrices, computer handling of large 29 definitions and conventions notation, purpose operations vii scalar product with 14 square symmetric vector product with 11 maximizing algorithm 181 meaningful boundedness 470 minors 68 minor of 70 mixed integer programming problem 637 mixed systems 205 parametric adjustment of 293 most negative element 152 multiplication by partitioning 22 instead of division 48 of matrices 10 Name codes 242 ordering convention 244 name lists 242 re-ordering conventions 244 negative definite matrix 341 roots 344 negative diagonal 420 pivot 437 negative definiteness 425 negative semidefinite matrix 341 roots 344 node index 67C nonadmissible restrictions 148 nonconvex mode operation 473, 474 problem 575, 600 programming problems 320 restriction 567, 641 CuuDuongThanCong.com 785 786 set 125 nonlinear duality 397 programming viii nonfeasible solution 144 nonnegativity restriction 486 nonstandard form block 433 tableau 433 non unique subsidiary optimum 599 normal transition under optimality 591 Objective function 144 limit 585 linear component, parametric variation of 525 parametric variation of 289 strictly convex 387 value, step length and 456 opening problem 273 operator 20, 62 optimal form condition 585 correction of 591, 601 loss of 236, 569 loss and correction of 594 subdominant 597 total 598 optimality 363 computational advantage of 236 condition 370 consistency and 588 without exhaustion 659 optimal solution 144 optimum solution 158 order parameter of a matrix outward adjustment 569 outward point 140, 142 overheads see fixed costs 639 overstatement of dual variable 595 by misidentification 595 simple 595 overtight approximation 564 Partitioning, multiplication by 22 of a vector 22 of matrices 21 parameter subspace, strict convexity in 530 parameter theorems 447 parameter treatment as variable 298 parametric equivalence 537 driving variable method 544 linear programming, computer implementation 306 methods in quadratic programming 516 CuuDuongThanCong.com INDEX INDEX Quasiconvexity see directional convexity Rank 68 full 92 of structured matrices 91 reapproximation 576 adjusted 583, 585 inwardly adjusted 583 reentry tableau for 579 recursive product 17 inversion of 56 references 773ff related problem 200 relation 116 restriction 124, 150 additive property of 363 nonconvex 641 nontrivial 124 polynomial, segmentation of 646 specified 638 revised simplex algorithm 267 explicit inverse without row updating 269 with row updating 267 product form inverse 270 row of a matrix operators, inversion by 84 permutation during inversion 51 Scalar matrix product with 14 secondary reentry column 720 second order condition, constrained 383 second order effect, displaced optimum segmentation of polynomial restrictions 646 semidefinite matrix factorization 356 sensitivity analysis 273 sequentially constrained maximization algorithm 560, 601 adaptations of 609-614 discussion 604-608 set, convex 124 of vectors 124 shadowprice of variable 159 sign inversion rule 418 simplex algorithm 144 for quadratic programming 410 outline 149 revised 267 see also revised simplex algorithm simplex step 156 tableau 151 matrix notation 160 CuuDuongThanCong.com 787 788 naming of rows and columns 242 nonupdated computerized 267 ordering of 244 packed storage of 267 printing 258 reordering convention 244 shortened 162 vector spaces and 171 singularity 33 slack variable 145 elimination 208 smallest quotient rule 153, 190, 250 conservative 190 solution vector 33 verification 574 specified restrictions 638 spuriously unbounded variable column 459 square matrix 19 standard form double step 466 steepest ascent principle 152, 197 step length and objective function value 456 strict convexity in parameter subspace 530 strictly anticonvex function 323 strictly convex function 323 quadratic 348 strict subspace convexity 406 suboptimality 659 suboptimal subbranch problem 673 subspace convexity viii, 383 subspace, Euclidean 122 linear 118 of definite matrices 350 substitute objective function 181 nonupdating of 202 substitution, algebraic, in matrix notation 15 subtraction of matrices subvector 22 sum count column 39 summation vector 62 superimposition of approximations 576, 580 blocked 582 full 582 unblocked 582 Tableau calculation of currently updated 577 larger subspace predecessor 433 neighbouring standard form 433 nonstandard form 436 and standard 433 CuuDuongThanCong.com INDEX INDEX 789 (nonstandard form) predecessor 441 normal sucessor 535 quadratic programming, ordering of 482 smaller subspace predecessor 433, 436 successor 433 use of tangential approximation 335 equivalent 336 of combined restrictions 365 subspace 334 theorems 22, 45, 46, 66, 70, 73, 75, 76, 77, 78, 79, 81, 92, 96, 98,108,119,121,125,126,141,173, 228ff.(Duality), 327, 338, 341, 345, 346, 350, 352, 356, 358, 365, 369 (Kuhn-Tucker), 390(Corner), 406, 408, 422, 425, 430, 432, 434(Nonstandard form block nonsingularity), 436(Smaller subspace immediate neighbour convexity transmission), 438,440, 442(complimentary pair), 446(Smaller subspace complementary pair transition), 449 (Driving variable parameter), 451 (Weak badname variable parameter), 453 (Primal badname convexity), 460, 470 (Convex primal boundedness), 586, 598, 721 topology of feasible space, definiteness and 319 transformed objective function 159 transpose of a matrix inversion of 56 triangular matrix 21 lower 21 upper 21 two-value columns 209 type absolute variable 145 Unbounded column 155 incoming variable 459 unbounded problem 155, 659 unit vector 20 un upda ting 719 upper bounds 205, 209 Van de Panne's algorithm 482, 558 variable, continuous 637 decision 638 dependant 116 dummy 641 explanatory 116 integer-restricted 637 leading 671 parameter treatment as 298 without nonnegativity restriction 486 without Sign restriction 205 zero-one 638 CuuDuongThanCong.com 790 vector 115, 118 column combination of 124 composite 22 matrix product with 11 parametric variation 274 parti tioning 22 permutation 64 proportionality 78 row solution 33 spaces, simplex tableau and 171 summation 62 unit 20 -vector multiplication verification of subsidiary optima 562 Weak badname variable parameter 451 Whinston/Cottle algorithm 558 Zero-one variable 638 mixed integer problem 656 CuuDuongThanCong.com INDEX ... Matrices and Simplex Algorithms A Textbook in Mathematical Programming and Its Associated Mathematical Topics D Reidel Publishing Company Dordrecht : Holland / Boston : U.S.A / London: England... of order m by n, and Ai and A' of order m1 by nand m2 by n Similarly, let B be of order n b~ k, and be partitioned into two block-columns B B1 and B2 being of order n by k1 and n by k CuuDuongThanCong.com... EQUATIONS-SYSTEMS AND TABLEAUX Chapter II/MATRIX NOTATION Chapter III/BLOCK-EQUATIONS AND MATRIX-INVERSION 33 Chapter IV / SOME OPERATORS AND THEIR USE 62 Chapter V / DETERMINANTS AND RANK 68 PART II GRAPHS AND
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