1040:
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657:
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56:
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441:
808:
82:
932:
223:
108:
587:
558:
481:
461:
384:
312:
197:
1569:
946:
1426:
801:
739:
23:
where the objective function and the constraint set can be biconvex. There are methods that can find the global optimum of these problems.
1507:
969:
563:
A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating
1598:
1021:
882:
989:
755:
Chen, Caihua (2016). ""The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent"".
1100:
794:
1562:
1593:
1377:
1039:
599:
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1105:
1421:
1389:
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659:
is block multi-convex iff it is convex with respect to each of the individual arguments while holding all others fixed.
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1095:
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994:
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817:
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29:
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1128:
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829:
20:
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951:
772:
707:
867:
61:
1223:
901:
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1303:
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202:
87:
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446:
369:
297:
182:
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711:
1403:
909:
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1527:
685:"Biconvex sets and optimization with biconvex functions: a survey and extensions"
589:
by fixing one of them and solving the corresponding convex optimization problem.
1490:
872:
768:
703:
731:
Deterministic global optimization : theory, methods, and applications
892:
1212:
592:
The generalization to functions of more than two arguments is called a
1447:
1263:
1126:
1054:
840:
790:
1038:
1543:
652:{\displaystyle f(x_{1},\ldots ,x_{K})\to \mathbb {R} }
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449:
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116:
90:
64:
32:
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217:
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102:
76:
50:
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802:
8:
281:
245:
166:
130:
692:Mathematical Methods of Operations Research
287:{\displaystyle B_{x}=\{y\in Y:(x,y)\in B\}}
172:{\displaystyle B_{y}=\{x\in X:(x,y)\in B\}}
1570:
1556:
1444:
1360:
1326:
1273:
1260:
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1123:
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787:
645:
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632:
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468:
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397:
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371:
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299:
236:
230:
204:
184:
121:
115:
89:
63:
31:
1043:Optimization computes maxima and minima.
674:
672:
366:is called a biconvex function if fixing
359:{\displaystyle f(x,y):B\to \mathbb {R} }
668:
1239:Principal pivoting algorithm of Lemke
7:
1524:
1522:
734:. Dordrecht : Kluwer Academic Publ.
883:Successive parabolic interpolation
51:{\displaystyle B\subset X\times Y}
14:
1203:Projective algorithm of Karmarkar
679:Gorski, Jochen; Pfeuffer, Frank;
1526:
1198:Ellipsoid algorithm of Khachiyan
1101:Sequential quadratic programming
938:Broyden–Fletcher–Goldfarb–Shanno
533:{\displaystyle f_{y}(x)=f(x,y)}
436:{\displaystyle f_{x}(y)=f(x,y)}
1156:Reduced gradient (Frank–Wolfe)
641:
638:
606:
527:
515:
506:
500:
430:
418:
409:
403:
348:
339:
327:
272:
260:
157:
145:
1:
1486:Spiral optimization algorithm
1106:Successive linear programming
1542:. You can help Knowledge by
1224:Simplex algorithm of Dantzig
1096:Augmented Lagrangian methods
58:is called a biconvex set on
1615:
1521:
1599:Applied mathematics stubs
1503:
1456:
1443:
1427:Push–relabel maximum flow
1272:
1259:
1229:Revised simplex algorithm
1135:
1122:
1063:
1050:
1036:
849:
836:
769:10.1007/s10107-014-0826-5
726:Floudas, Christodoulos A.
704:10.1007/s00186-007-0161-1
77:{\displaystyle X\times Y}
952:Symmetric rank-one (SR1)
933:Berndt–Hall–Hall–Hausman
757:Mathematical Programming
1476:Parallel metaheuristics
1284:Approximation algorithm
995:Powell's dog leg method
947:Davidon–Fletcher–Powell
843:Unconstrained nonlinear
19:is a generalization of
1538:-related article is a
1461:Evolutionary algorithm
1044:
653:
583:
554:
534:
477:
457:
437:
380:
360:
308:
288:
219:
218:{\displaystyle x\in X}
193:
173:
104:
103:{\displaystyle y\in Y}
78:
52:
1594:Generalized convexity
1234:Criss-cross algorithm
1057:Constrained nonlinear
1042:
863:Golden-section search
654:
596:function. A function
584:
555:
535:
478:
458:
438:
381:
361:
309:
289:
220:
194:
174:
105:
79:
53:
17:Biconvex optimization
1151:Cutting-plane method
600:
567:
544:
487:
467:
447:
390:
370:
321:
298:
229:
203:
199:and for every fixed
183:
114:
88:
62:
30:
1589:Convex optimization
1536:applied mathematics
1481:Simulated annealing
1299:Integer programming
1289:Dynamic programming
1129:Convex optimization
990:Levenberg–Marquardt
582:{\displaystyle x,y}
294:is a convex set in
179:is a convex set in
84:if for every fixed
21:convex optimization
1161:Subgradient method
1045:
970:Conjugate gradient
878:Nelder–Mead method
649:
594:block multi-convex
579:
550:
530:
473:
453:
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100:
74:
48:
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1034:
1033:
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1029:
1008:
1007:
741:978-0-7923-6014-8
681:Klamroth, Kathrin
553:{\displaystyle X}
476:{\displaystyle y}
456:{\displaystyle Y}
379:{\displaystyle x}
307:{\displaystyle Y}
192:{\displaystyle X}
1606:
1572:
1565:
1558:
1530:
1523:
1445:
1361:
1327:
1304:Branch and bound
1294:Greedy algorithm
1274:
1261:
1181:
1137:
1124:
1065:
1052:
1000:Truncated Newton
915:Wolfe conditions
898:
851:
838:
811:
804:
797:
788:
781:
780:
752:
746:
745:
722:
716:
715:
689:
683:(22 June 2007).
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49:
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1314:
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1174:
1165:
1142:
1131:
1110:
1084:
1080:Penalty methods
1075:Barrier methods
1059:
1046:
1026:
1022:Newton's method
1004:
956:
919:
887:
868:Powell's method
845:
832:
815:
785:
784:
754:
753:
749:
742:
724:
723:
719:
687:
678:
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628:
609:
598:
597:
565:
564:
542:
541:
540:is convex over
490:
485:
484:
465:
464:
445:
444:
443:is convex over
393:
388:
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368:
367:
319:
318:
296:
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201:
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117:
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28:
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12:
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5:
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1450:Metaheuristics
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1437:
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1424:
1422:Ford–Fulkerson
1419:
1414:
1408:
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1400:
1399:
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1395:
1393:
1392:
1390:Floyd–Warshall
1387:
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1236:
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1226:
1220:
1218:
1209:
1208:
1206:
1205:
1200:
1195:
1193:Affine scaling
1189:
1187:
1185:Interior point
1178:
1167:
1166:
1164:
1163:
1158:
1153:
1147:
1145:
1133:
1132:
1127:
1120:
1119:
1116:
1115:
1112:
1111:
1109:
1108:
1103:
1098:
1092:
1090:
1089:Differentiable
1086:
1085:
1083:
1082:
1077:
1071:
1069:
1061:
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1055:
1048:
1047:
1037:
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1024:
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865:
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847:
846:
841:
834:
833:
816:
814:
813:
806:
799:
791:
783:
782:
763:(1–2): 57–59.
747:
740:
717:
698:(3): 373–407.
667:
666:
664:
661:
647:
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635:
631:
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621:
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612:
608:
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549:
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38:
35:
13:
10:
9:
6:
4:
3:
2:
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1492:
1489:
1487:
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1469:
1467:
1466:Hill climbing
1464:
1462:
1459:
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1455:
1451:
1446:
1442:
1428:
1425:
1423:
1420:
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1410:
1409:
1407:
1405:
1404:Network flows
1401:
1391:
1388:
1386:
1383:
1379:
1376:
1375:
1374:
1371:
1370:
1368:
1366:
1365:Shortest path
1362:
1352:
1349:
1347:
1344:
1342:
1339:
1338:
1336:
1334:
1333:spanning tree
1328:
1325:
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1305:
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1301:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1281:
1279:
1275:
1271:
1267:
1266:Combinatorial
1262:
1258:
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1235:
1232:
1230:
1227:
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1199:
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1041:
1023:
1020:
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1015:
1011:
1001:
998:
996:
993:
991:
988:
986:
983:
981:
978:
976:
973:
971:
968:
967:
965:
963:
962:Other methods
959:
953:
950:
948:
945:
943:
939:
936:
934:
931:
930:
928:
926:
922:
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903:
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848:
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789:
778:
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733:
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727:
721:
718:
713:
709:
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697:
693:
686:
682:
675:
673:
669:
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660:
633:
629:
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622:
619:
614:
610:
603:
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576:
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561:
547:
524:
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503:
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491:
470:
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427:
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412:
406:
398:
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373:
345:
342:
336:
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324:
315:
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278:
275:
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186:
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133:
127:
122:
118:
97:
94:
91:
71:
68:
65:
45:
42:
39:
36:
33:
24:
22:
18:
1544:expanding it
1533:
1518:
1471:Local search
1417:Edmonds–Karp
1373:Bellman–Ford
1143:minimization
975:Gauss–Newton
925:Quasi–Newton
910:Trust region
818:Optimization
760:
756:
750:
730:
720:
695:
691:
593:
591:
562:
316:
25:
16:
15:
1491:Tabu search
902:Convergence
873:Line search
463:and fixing
317:A function
1583:Categories
1322:algorithms
830:heuristics
822:Algorithms
663:References
1277:Paradigms
1176:quadratic
893:Gradients
855:Functions
642:→
623:…
349:→
276:∈
252:∈
210:∈
161:∈
137:∈
95:∈
69:×
43:×
37:⊂
1508:Software
1385:Dijkstra
1216:exchange
1014:Hessians
980:Gradient
728:(2000).
712:15900842
1351:Kruskal
1341:BorĹŻvka
1331:Minimum
1068:General
826:methods
777:5646309
1213:Basis-
1171:Linear
1141:Convex
985:Mirror
942:L-BFGS
828:, and
775:
738:
710:
26:A set
1534:This
1412:Dinic
1320:Graph
773:S2CID
708:S2CID
688:(PDF)
1540:stub
1378:SPFA
1346:Prim
940:and
736:ISBN
1308:cut
1173:and
765:doi
761:155
700:doi
1585::
824:,
820::
771:.
759:.
706:.
696:66
694:.
690:.
671:^
560:.
483:,
386:,
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110:,
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1546:.
1306:/
810:e
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767::
744:.
714:.
702::
646:R
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634:K
630:x
626:,
620:,
615:1
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604:f
577:y
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525:y
522:,
519:x
516:(
513:f
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507:)
504:x
501:(
496:y
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428:y
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416:f
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399:x
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282:}
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270:y
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264:x
261:(
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