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Index

A-konjugiert , 149

Phase I der Simplexmethode, 44

Abl eitung- Frechetsche - , 164- Gäteaux'sche - , 164- höh ere - , 164Ab stiegsmethoden, 135, 139Accessibili ty Lemma, 213Adjazenzmatrix , 104affin invariant , 170, 362affine Hülle, 207ak tiv , 49- Ungleichung , 9, 29an alytisches Zen trum, 76, 358Armijo line search , 145Augmented Lagr angian , 299Au sgleichsproblem- nichtlineares, 184

Barri erefunktion , 76, 316Barrieremethode, 318Barrieremethoden, 315Basis, 23- zulässige, 25Basislösung, 25Basisvari able, 23Baum, 107Bedingung 1. Ordnung- (notwendige -) , 249Bedingung 2.0rdnung- hinreichende - , 260- notwendige - , 258BFGS-Verfahren, 180bip artit- -er Gr aph, 428box-constraints, 273

Broydensche ß-Klasse, 182

Cau chy-Schwarz 'sehe Un gleichung- verallgem einer te, 396- verallgem einer te - , 367cg-Verfah ren , 148, 150Cholesky-Zerlegung, 78, 152chromat ische Zahl- eines Gr aphen , 427Clique- maximale, 427cons traint qu alifica tion- 2. Ordnung, 260- Fritz John , 229- LICQ, 253- von Slater , 228CPM , 121

Dantzig- Methode von , 117DFP-Verfahren , 181Dijkstra- Methode von , 119Direkte Suchverfahren , 129direk tes Verfahren , 152Diätprobl em , 10Dr eieckszerlegungsm ethode, 47dual- - er Simplexschri t t , 55- zulässige Basi s, 55dualer Kegel, 216Du alität- - ssa tz der linear en Op timierung, 52,

54bei konvexen P rogrammen inkoni scher Form, 235

Du alität slücke , 75dünn besetzt , 13, 92

472 Index

Ecke, 19Eigenwertoptimierung, 419Ellipse- äußere - , 371, 375- Innere, 364entartet , 29Extremalmenge, 19Extremalpunkt , 19

Farkas Lemma, 65Fehlerquadrats umme, 185Filte r-Verfahren, 349Finsler- Lemma von - , 301freie Variabl e, 14, 26Frit z John constraint qu alification , 229Fritz-John-Bedingung, 229Frobeniusmat rix, 33, 39Frechet-differenzierbar , 164, 196Fulkerson- Methode von , 120Färbung- eines Graphen , 427Fejer- Sat z von , 238, 403

Gauß-Newton-Verfah ren , 187Givensro tation, 48glob ales Minimum, 127Goeman s-Willi am son- Verfahren, 439gold ener Schnitt- Verfah ren des - , 130Graph,103Graphenparti t ioni erung, 442Gr aphentheorie , 101Gäte au x-differenzierbar , 164, 196

H-Norm, 364Hessematrix, 128Hir sch- Vermutung von - , 47Höldersche Un gleichung, 312

Indexvektor, 23induzier ter Graph, 106Innere-Punkte-Methoden- für konvexe P rogramme, 355- für lineare Programme, 67Innere-Punkte-Verfahren

- un zulässiges, 386, 388Inverse-Basis-Methode, 47Inz idenzmatrix , 103

J acobi-Matrix, 73

Kan te, 20, 103Karush- Sat z von - .Kuhn und Tucker , 223Kegel, 49, 204- duale, 216- pol are , 216KK T-Bedingungen, 230Klee-Minty-Probleme, 65Knoten , 103Komplem entari tä t , 50Komplementgraph, 427komplementär- Indexvektor, 23- st r ikt, 262koni sche Form- P roblem in - , 51- von konvexen P rogrammen , 233Konjugier te Gr adienten-Verfahren, 148konkav- - e Funktion , 218Konvergenzraten , 68konve x- -e Funktion , 16, 218- -e Menge, 16, 204- - e quadra t ische Funktion , 146- streng - , 16, 76- st reng -e Funktion, 218konvexe Hülle, 205kreisfreier Graph, 120Kriging-Verfah ren , 456Kulm und Tucker- Sat z von Karush , - , 223Kuhn-Tucker Punkt , 249Kurz-S chri t t-Algorithmus, 80kün stliche Vari able, 44

Lagrangedualität , 421, 424, 433Lagran gefunktion, 231, 245- erweiterte - , 299Lagran gemultiplikator, 249least-square s-Problem- nichtlineares, 184lexikoposit iv , 41

line search, 129- Armijo, 145- exakte - , 145, 150linear- - e Konvergenz , 71line arisierter Kegel, 246Lipschitzbedingung- relative, 414lokale s Minimum, 127Lovasz-Zahl, 432Lyapunovungleichung, 417Lösung- Optimal-, 2- zul ässige, 2, 14Löwner- Halbordnung, 380

Ma ratos-Effekt , 336, 349Max-Cut P roblem , 434meri t fun ction, 333Methode der Zentren , 357Minkowski-Funktional , 414monotoner Op era to r , 275

NC P-Fu nkt ion, 325Nelder-Mead-Verfahren, 453Netzwer k, 101Newt on-Kanto rovi ch- Satz von - , 163Newt on-Verfahren , 68, 368 , 369, 383- Minimier ung mi t nichtlinearen

Gleichungsrestriktionen , 322Nicht basis, 24Nicht basisvariable, 24nichtentartet , 29- du al - , 56Niveaume nge, 143Norm- lubx, , 212- lub2-, 142- Frobenius- , 142Normalgleichunge n, 185NW-Eckenregel, 109

O-Notation, 71Op timali tä tsb edingung- für allgemei ne Optimierungsproble­

me , 243- für das Transshipmentproblem , 115

Index 473

- für konvexe Optimierungsprobleme,225

Oren-Luenberger-Kl asse, 182Orthogonalproj ek tion, 78- auf eine konvexe Menge, 241Orthonormalbasis, 179

Pen alty- Funktion, 293- differenzierbare - , 298- exakte - , 296PERT, 121Pivotelem en t , 36Pol ak -Ribiere- Verfahren von - , 154po larer Kegel , 216Polyed er , 16po lynomiale Laufz eit , 67, 87, 91Pol ytop- Max-Cut-, 436- metrisches, 439- stabile-Mengen-, 429Powell- Updateforme l von , 309P redikto r-Korr ek to r-Verfahren- primales - , 389Prim al- -duales Innere-Punkte-Verfahren , 319profi table Richtung, 279Programm- lineares, 9- nichtlineares, 2Projek tion- auf konvexe Menge, 275Projektionsverfahren, 273proj izierter Gradient , 279Prädiktor-Korrektor-Verfahren- von Mehrotra , 88Präkonditionierung. 153PSB-Verfahren, 190

Q-quadra tis ch- - e Konvergenz, 71Quasi-Newton- - Bedingung, 176- - Verfahren , 173, 176, 189

R-quadratisch- - e Konvergenz, 71Rang-1-Verfahren

474 Index

- von Broyden, 176reduzierte Kosten, 34Regular itätsbedingung- 2. Ordnung, 260- Fritz John , 229- von Robinson , 248- von Sla ter , 228rel ati v innere Punkte, 211Rel ative Lipschi t z-Bedingung, 365relativer Randpunkt , 211Relaxi erung- semidefinit e, 422, 433 , 440Residuum, 77, 79rezessiver Kegel , 397Robinson- Regularitätsbedingung, 248Ros enbrock-Funkt ion- vera llgemeinert e - , 139

Sattelpunkt , 231Scha t tenpreise. 31, 61schiefsym met rische Matrix, 93Schlupfvari able, 14Schurkomplem ent , 420second ord er correct ion , 336Selbstbeschränkung, 372selbst duales line ares Programm, 93Selb stkonkordanz , 359, 360sem idefinit e P rogramme, 237 , 363- nichtlineare, 447sem idefinites P rogramm, 403Sem idefinitheitsb edingung, 363Sen siti vit ät sanalyse- bei linearen Programmen, 58- bei nichtlinearen P rogrammen , 266Sequenti al Quadrati c Programs, 327Sherman-Morrison-vVoodbury-Formel ,

61, 307Shifted Penalty Mul t ipli er Method , 309Simplex , 453Simplexform, 26- allgeme inere- , 40Simplexmethode- lexikographische, 41- Name der - , 453- von Neider und Mead, 453Simplexschrit t , 36Skalierungsinvarianz, 415Sp okt ral radius, 188

Spi ralfunkt ion , 138SQP-Verfahren , 327, 449st abil- Differentialgleichung, 417stabile Menge- in einem Graphen , 427St andardform- eines linearen Programmes, 14- eines linearen P rogrammes, 13stationärer Punkt , 128, 274, 341- sing ulärer , 342- un zul ässiger, 342steilster Abstieg- Kurve des - , 136- Verfahren des - , 146Straffunktion , 293st rikt komplem entäre Lösung, 93, 100,

264Subgradien t , 221Suchrichtung- AHO , 408 , 412 , 416- HKM, 408, 412- Klasse MZ , 408- NT , 408 , 413superl ineare Konvergenz, 174Symmetrisierung- bei semidefinit en P rogrammen , 408

Tableau , 25Tangentialkegel, 243Taylor- Satz von , 68Tr ansportpr oblem , 101Transsh ipment-Problem , 113Tr ennung , 208- eigentliche, 208- strikte, 208Tr ennungss atz, 203Trilinear form, 68- Spektralradius von symmet r ischer - ,

367Tr ust-Regi on Verfahren , 155Trust-Region-Verfahren- bei Nebe nbeding ungen , 339, 340

unimodal- -e Funktion , 129un imodulare Matrix , 113, 122universale Barrierefunkt ion , 374

unterhalbstetig, 281

Weg , 106- kürzester, 117- läng ster, 117- ungerichtet , 106

zentraler

- Pfad , 406zent ra ler Pfad , 74Zielfunktion, 2Zoutendijk , 338zuläss ige Richtung, 279zusammenhängend, 106Zwischenwert , 69Zyklus, 106

Index 475