REFERENCES

[1] Abadie, J. (Editor) Nonlinear Programming Amsterdam, North Holland Publishing Co., 1967 Referred to: This bibliography [3J, [40].

(2) Baumol, W.J. Economic Theory and Operations Analysis Prentice Hall, !-.IewJersey, U.S.A., 1951, 1965, 1972 Referred to: 11.2

[3] Beale, E.M.L. An Introduction to Beale's In: Abadie (Editor) [1], Referred to: 16.11

[4] Charnes, A.

Method of Quadratic Programming 1967, pp. 143-150

Optimality and Degeneracy in Linear Programming. Econometrica, 20 (1952), pp. 160-170 Referred to: 8.10

[5] Charnes, A., Cooper, W.W., and Henderson, A. An Introduction to Linear Programming. New York, John Wiley, 1953, 1958 Referred to: 7.1, 8.10

[6] Cottle, R.W. The Principal Pivoting Method of Quadratic Programming. In: Dantzig, G. B., and Veinott, A. F. (Editors) Mathematics of the Decision Sciences. (Proceedings of the 1967 Summer School) American Mathematical Society, 1968 Referred to: 16.2, 16.3

773

774

[7] Dakin, R.J. A Tree-Search Algorithm for Mixed Integer Programming

Problems. The Computer Journal, 8 (Apr 1965), pp.250-255 Referred to: 20.2, 20.3

[8] Dantzig, G.B. Linear Programming and Extensions. Princeton University Press, Princeton, U.S.A., 1963 Referred to: 11.3,13.1,16.2

[9] Dantzig, G., Eisenberg, A. and Cottle, R.W. Symmetric Dual Nonlinear Programs Operations Research Center, University of California Researcg Report No. 38 Subsequently published in: Pacific Journal of Mathematics ~, 1965, pp.809-8l2

[101 Fiacco, A.V., and McCormick, r..P. Non-Linear Programming; Sequential Unconstrained

Maximization Techniques. New York, Wiley, 1968. Referred toc 18.2

[11] Garvin, W.W. Introduction to Linear Programming. New York, McGraw Hill, 1960 Referred to: 8.1,9.5,11.2,13.1

[12] Gass, S.l., and Saati, T. Parametric Objective Function.

REFERENCES

Journal of t.he Operations Research S-ociety of America, 2 (l9541, pp.3l6-3l9

Referred to: 13.1

[13] Gass, S.l., and Saati, T. The Computadonal Algorithm for the Parametric Objective

Function. Naval Research Logistics. Quarterly, 2 (19551, pp. 39."."5 Referred to! 13.1

[14] Geary, R.C. Elements of Linear Programmin;g.With Economic Applications Griffins Statistical Monographs and Courses Nr. 15 Griffin, London 1973 Referred to: 8,10

REFERENCES

[15] Gomory, R. An Algorithm for Integer Solutions to Linear Problems In: Graves, R.L., and Wolfe, P. (Editors) Recent Advances in Mathematical Programming. McGraw Hill, 1963 Referred to: 21.1

[16] Gomory, R. An Algorithm for the Mixed Integer Problem. Rand Corporation p. 1885, Santa Monica, Cal. U.S.A.,

June 1960 Referred to: 21.1

[17] Goncalves, A. Primal-Dual and Parametric Methods in Mathematical

Programming. Thesis, Ph.D., Birmingham, Dept. of Mathematical Statistics,

1970. Referred to: 17.6

[18] Heesterman, A.R.G. Allocation Models and their Use in Economic Planning. Reide 1, Dordrecht, Netherlands, 1971 Referred to: 8.10

[19] Heesterman, A.R.G. Forecasting Models for National Economic Planning. Reidel, Dordrecht, Netherlands, 1970, 1972 Referred to: 4.1

[20] Heesterman, A.R.G. Special Simplex Algorithm for Multi-Sector Problems. Numerische Mathematik 12 (1968) pp. 288-306 (In English) Referred to: 8.2, l3.Z-

[21] Hahn, F. Elementary Matrix Algebra. The Macmillan Company, New York, & Collier Macmillan, London, U.K., and Toronto, Canada 1973 Referred to: 14.4

[22] John, F. Extremum Properties with Inequalities as Subsidiary

Condi tions. (Presented to R. Courant) New York, Interscience Publishers, 1968. Referred to: 15.3

775

776

[23] Kim, C. Introduction to Linear Programming. New York, Rinehart and Winston, 1971 Referred to: 11.3

REFERENCES

[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

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. Foundation~ of Economic Analysis. Harvard University Press, Cambridge, Mass., U.S.A., 1967 Referred to: 15.5

[ 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, 2 (1964) pp. 2]3-302 Referred to: 16.3

777

77 8 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

I N D E X

Accuracy 166 addition of matrices 9

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

780

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 5 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

INDEX

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

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 7 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

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

INDEX

element of a matrix 5 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 3 examples 10, 11, 12, 13, 14, 16, 17, 19, 20, 21,

34, 35, 37, 38, 40, 43, 44, 47, 48, 49, 51, 22, 23, 24, 55, 56, 59,

66, 68, 73, 82, 83, 85, 94, 95, 96, 103, 111, 117, 118, 123, 127, 130, 134, 138, 145, 154, 165, 177, 179, 182, 189, 202, 205, 208, 209, 210, 217, 218, 231, 232, 235, 268, 269, 270, 271, 276, 287, 290, 293, 295, 304, 319, 322, 332, 335, 340, 343, 344, 349, 353, 354, 370, 374, 379, 383, 386, 392, 395, 404, 408, 411, 429, 439, 443, 459, 462, 468, 476, 480, 487, 489, 492, 495, 526, 531, 542, 546, 556, 561, 563, 567, 568, 574, 576, 583, 591, 595, 599, 600, 612, 637, 639, 640, 641, 669, 703, 711, 739

783

27, 62, 120,

188, 240, 321, 375, 4S8, 533, 594, 713,

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

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

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 5 notation, purpose 5 operations vii scalar product with 14 square 6 symmetric 7 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

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 6 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

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 5 operators, inversion by 84 permutation during inversion 51

Scalar 7 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

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 9 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

INDEX

INDEX

(nonstandard form) predecessor 441

normal sucessor 535 quadratic programming, ordering of 482 smaller subspace predecessor 433, 436

successor 433 use of 4

tangential approximation 335 equivalent 336

of combined restrictions 365 subspace 334

789

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 8

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

790

vector 115, 118 column 6 combination of 124 composite 22 matrix product with 11 parametric variation 274 parti tioning 22 permutation 64 proportionality 78 row 6 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

INDEX