344:
774:, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.
153:
339:{\displaystyle {\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\\&&&h_{j}(x)=0,\quad j=1,\dots ,p\end{aligned}}}
719:
958:
158:
951:
838: â optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints
944:
1148:
1048:
621:
1140:
898:
873:
759:
that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from
1153:
794:
823: â Cognitive heuristic of searching for an acceptable decision â the optimum need not be found, just a "good enough" solution.
1173:
1387:
1090:
783:
737:
107:
1158:
477:
471:
1188:
835:
31:
1178:
1163:
1005:
396:
147:
128:
1248:
1225:
1119:
1043:
771:
119:
1366:
1327:
1243:
1168:
1095:
1080:
1033:
917:
578:
73:
1100:
598:
592:
570:
90:
58:
1105:
971:
797: â theorem that asserts that there exist nearly optimal solutions to some optimization problems
1038:
1028:
1023:
814:
789:
450:, the problem is an unconstrained optimization problem. By convention, the standard form defines a
124:
95:
77:
1267:
985:
860:
365:
1085:
894:
869:
512:
81:
66:
1238:
1183:
1074:
1069:
803:
724:
459:
46:
936:
1258:
1229:
1203:
1198:
1193:
1124:
1109:
1018:
990:
967:
355:
767:
that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
1345:
1253:
1114:
1013:
826:
142:
1381:
1312:
1304:
1300:
1296:
1292:
1288:
1129:
809:
111:
1350:
359:
72:
Optimization problems can be divided into two categories, depending on whether the
1340:
1335:
1219:
820:
581:
103:
42:
38:
17:
1263:
1233:
995:
817: â Discipline concerning the application of advanced analytical methods
50:
727:
that asks whether there is a feasible solution for some particular measure
1272:
1053:
99:
829: â type of computational problem represented by a binary relation
723:
For each combinatorial optimization problem, there is a corresponding
940:
88:
An optimization problem with discrete variables is known as a
714:{\displaystyle m(x,y)=g\left\{m(x,y'):y'\in f(x)\right\}.}
751:, an optimization problem might be "find a path from
624:
156:
840:
Pages displaying wikidata descriptions as a fallback
831:
Pages displaying wikidata descriptions as a fallback
799:
Pages displaying wikidata descriptions as a fallback
1359:
1326:
1281:
1212:
1138:
1062:
1004:
978:
713:
338:
117:A problem with continuous variables is known as a
918:"How Traffic Shaping Optimizes Network Bandwidth"
859:Boyd, Stephen P.; Vandenberghe, Lieven (2004).
952:
27:Problem of finding the best feasible solution
8:
868:. Cambridge University Press. p. 129.
606:The goal is then to find for some instance
959:
945:
937:
214:
623:
286:
229:
192:
162:
157:
155:
30:For broader coverage of this topic, see
1049:Locally convex topological vector space
889:Ausiello, Giorgio; et al. (2003),
851:
806: â Type of computational problem
786: â Type of computational problem
7:
590:is the goal function, and is either
123:, in which an optimal value from a
466:Combinatorial optimization problem
218:
215:
211:
208:
205:
202:
199:
196:
193:
25:
539:is the set of feasible solutions;
893:(Corrected ed.), Springer,
127:must be found. They can include
1154:Ekeland's variational principle
795:Ekeland's variational principle
614:, that is, a feasible solution
310:
253:
136:Continuous optimization problem
700:
694:
674:
657:
640:
628:
298:
292:
241:
235:
184:
178:
1:
784:Counting problem (complexity)
736:. For example, if there is a
891:Complexity and Approximation
1174:HermiteâHadamard inequality
1404:
478:combinatorial optimization
472:Combinatorial optimization
469:
29:
836:Semi-infinite programming
369:to be minimized over the
32:Mathematical optimization
1360:Applications and related
1164:Fenchel-Young inequality
772:approximation algorithms
743:which contains vertices
546:and a feasible solution
462:the objective function.
150:optimization problem is
131:and multimodal problems.
1120:Legendre transformation
1044:Legendre transformation
120:continuous optimization
1388:Computational problems
1367:Convexity in economics
1301:(lower) ideally convex
1159:FenchelâMoreau theorem
1149:Carathéodory's theorem
715:
340:
1289:Convex series related
1189:ShapleyâFolkman lemma
716:
577:, which is usually a
341:
110:must be found from a
91:discrete optimization
1179:KreinâMilman theorem
972:variational analysis
622:
456:maximization problem
452:minimization problem
417:equality constraints
154:
129:constrained problems
55:optimization problem
1169:Jensen's inequality
1039:Lagrange multiplier
1029:Convex optimization
1024:Convex metric space
862:Convex Optimization
815:Operations research
790:Design Optimization
125:continuous function
1297:(cs, bcs)-complete
1268:Algebraic interior
986:Convex combination
711:
542:given an instance
518:given an instance
458:can be treated by
366:objective function
336:
334:
170:
67:feasible solutions
65:solution from all
1375:
1374:
900:978-3-540-65431-5
875:978-0-521-83378-3
373:-variable vector
163:
16:(Redirected from
1395:
1293:(cs, lcs)-closed
1239:Effective domain
1194:RobinsonâUrsescu
1070:Convex conjugate
961:
954:
947:
938:
933:
931:
929:
904:
903:
886:
880:
879:
867:
856:
841:
832:
804:Function problem
800:
770:In the field of
766:
762:
758:
754:
750:
746:
742:
735:
725:decision problem
720:
718:
717:
712:
707:
703:
687:
673:
617:
612:optimal solution
609:
601:
595:
589:
576:
568:
553:
549:
545:
538:
527:
510:
503:
483:
449:
434:
427:
414:
392:
376:
372:
362:
345:
343:
342:
337:
335:
291:
290:
280:
279:
278:
234:
233:
223:
221:
190:
173:
171:
160:
47:computer science
21:
18:Optimal solution
1403:
1402:
1398:
1397:
1396:
1394:
1393:
1392:
1378:
1377:
1376:
1371:
1355:
1322:
1277:
1208:
1134:
1125:Semi-continuity
1110:Convex function
1091:Logarithmically
1058:
1019:Convex geometry
1000:
991:Convex function
974:
968:Convex analysis
965:
927:
925:
916:
913:
908:
907:
901:
888:
887:
883:
876:
865:
858:
857:
853:
848:
839:
830:
798:
780:
764:
760:
756:
752:
748:
744:
740:
734:
728:
680:
666:
653:
649:
620:
619:
615:
607:
597:
591:
587:
574:
555:
551:
547:
543:
529:
519:
508:
485:
484:is a quadruple
481:
474:
468:
440:
429:
422:
407:
402:
385:
380:
374:
370:
350:
333:
332:
282:
276:
275:
225:
222:
188:
187:
172:
152:
151:
138:
61:of finding the
35:
28:
23:
22:
15:
12:
11:
5:
1401:
1399:
1391:
1390:
1380:
1379:
1373:
1372:
1370:
1369:
1363:
1361:
1357:
1356:
1354:
1353:
1348:
1346:Strong duality
1343:
1338:
1332:
1330:
1324:
1323:
1321:
1320:
1285:
1283:
1279:
1278:
1276:
1275:
1270:
1261:
1256:
1254:John ellipsoid
1251:
1246:
1241:
1236:
1222:
1216:
1214:
1210:
1209:
1207:
1206:
1201:
1196:
1191:
1186:
1181:
1176:
1171:
1166:
1161:
1156:
1151:
1145:
1143:
1141:results (list)
1136:
1135:
1133:
1132:
1127:
1122:
1117:
1115:Invex function
1112:
1103:
1098:
1093:
1088:
1083:
1077:
1072:
1066:
1064:
1060:
1059:
1057:
1056:
1051:
1046:
1041:
1036:
1031:
1026:
1021:
1016:
1014:Choquet theory
1010:
1008:
1002:
1001:
999:
998:
993:
988:
982:
980:
979:Basic concepts
976:
975:
966:
964:
963:
956:
949:
941:
935:
934:
924:. 12 July 2016
912:
911:External links
909:
906:
905:
899:
881:
874:
850:
849:
847:
844:
843:
842:
833:
827:Search problem
824:
818:
812:
807:
801:
792:
787:
779:
776:
732:
710:
706:
702:
699:
696:
693:
690:
686:
683:
679:
676:
672:
669:
665:
662:
659:
656:
652:
648:
645:
642:
639:
636:
633:
630:
627:
604:
603:
585:
540:
516:
470:Main article:
467:
464:
437:
436:
420:
405:
400:
383:
378:
331:
328:
325:
322:
319:
316:
313:
309:
306:
303:
300:
297:
294:
289:
285:
281:
277:
274:
271:
268:
265:
262:
259:
256:
252:
249:
246:
243:
240:
237:
232:
228:
224:
220:
217:
213:
210:
207:
204:
201:
198:
195:
191:
189:
186:
183:
180:
177:
174:
169:
166:
161:
159:
137:
134:
133:
132:
115:
94:, in which an
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1400:
1389:
1386:
1385:
1383:
1368:
1365:
1364:
1362:
1358:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1333:
1331:
1329:
1325:
1318:
1316:
1310:
1308:
1302:
1298:
1294:
1290:
1287:
1286:
1284:
1280:
1274:
1271:
1269:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1231:
1227:
1223:
1221:
1218:
1217:
1215:
1211:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1184:Mazur's lemma
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1146:
1144:
1142:
1137:
1131:
1130:Subderivative
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1078:
1076:
1073:
1071:
1068:
1067:
1065:
1061:
1055:
1052:
1050:
1047:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1011:
1009:
1007:
1006:Topics (list)
1003:
997:
994:
992:
989:
987:
984:
983:
981:
977:
973:
969:
962:
957:
955:
950:
948:
943:
942:
939:
923:
919:
915:
914:
910:
902:
896:
892:
885:
882:
877:
871:
864:
863:
855:
852:
845:
837:
834:
828:
825:
822:
819:
816:
813:
811:
810:Glove problem
808:
805:
802:
796:
793:
791:
788:
785:
782:
781:
777:
775:
773:
768:
739:
731:
726:
721:
708:
704:
697:
691:
688:
684:
681:
677:
670:
667:
663:
660:
654:
650:
646:
643:
637:
634:
631:
625:
613:
600:
594:
586:
583:
580:
572:
566:
562:
558:
541:
536:
532:
526:
522:
517:
515:of instances;
514:
507:
506:
505:
501:
497:
493:
489:
479:
473:
465:
463:
461:
457:
453:
447:
443:
432:
425:
421:
418:
412:
408:
401:
399:
398:
390:
386:
379:
368:
367:
361:
357:
353:
349:
348:
347:
329:
326:
323:
320:
317:
314:
311:
307:
304:
301:
295:
287:
283:
272:
269:
266:
263:
260:
257:
254:
250:
247:
244:
238:
230:
226:
181:
175:
167:
164:
149:
145:
144:
143:standard form
135:
130:
126:
122:
121:
116:
113:
112:countable set
109:
105:
101:
97:
93:
92:
87:
86:
85:
83:
79:
75:
70:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
1351:Weak duality
1314:
1306:
1226:Orthogonally
926:. Retrieved
921:
890:
884:
861:
854:
769:
729:
722:
611:
605:
569:denotes the
564:
560:
556:
534:
530:
524:
520:
499:
495:
491:
487:
476:Formally, a
475:
455:
451:
445:
441:
438:
430:
423:
416:
410:
403:
394:
388:
381:
364:
351:
141:
139:
118:
89:
71:
62:
54:
36:
1341:Duality gap
1336:Dual system
1220:Convex hull
928:13 February
821:Satisficing
415:are called
397:constraints
395:inequality
393:are called
104:permutation
98:such as an
43:engineering
39:mathematics
1264:Radial set
1234:Convex set
996:Convex set
846:References
148:continuous
78:continuous
1249:Hypograph
689:∈
324:…
267:…
245:≤
74:variables
51:economics
1382:Category
1273:Zonotope
1244:Epigraph
778:See also
685:′
671:′
579:positive
504:, where
480:problem
460:negating
354: :
165:minimize
82:discrete
1328:Duality
1230:Pseudo-
1204:Ursescu
1101:Pseudo-
1075:Concave
1054:Simplex
1034:Duality
571:measure
363:is the
100:integer
59:problem
57:is the
1311:, and
1282:Series
1199:Simons
1106:Quasi-
1096:Proper
1081:Closed
897:
872:
346:where
96:object
1139:Main
866:(pdf)
738:graph
618:with
511:is a
419:, and
413:) = 0
391:) †0
146:of a
108:graph
53:, an
1259:Lens
1213:Sets
1063:Maps
970:and
930:2017
895:ISBN
870:ISBN
747:and
582:real
454:. A
428:and
140:The
76:are
63:best
49:and
1313:(Hw
922:IPC
763:to
755:to
610:an
599:max
596:or
593:min
573:of
550:of
513:set
448:= 0
439:If
433:â„ 0
426:â„ 0
106:or
84::
80:or
37:In
1384::
1305:(H
1303:,
1299:,
1295:,
1232:)
1228:,
1108:)
1086:K-
920:.
563:,
554:,
528:,
523:â
498:,
494:,
490:,
444:=
358:â
102:,
69:.
45:,
41:,
1319:)
1317:)
1315:x
1309:)
1307:x
1291:(
1266:/
1224:(
1079:(
960:e
953:t
946:v
932:.
878:.
765:v
761:u
757:v
753:u
749:v
745:u
741:G
733:0
730:m
709:.
705:}
701:)
698:x
695:(
692:f
682:y
678::
675:)
668:y
664:,
661:x
658:(
655:m
651:{
647:g
644:=
641:)
638:y
635:,
632:x
629:(
626:m
616:y
608:x
602:.
588:g
584:.
575:y
567:)
565:y
561:x
559:(
557:m
552:x
548:y
544:x
537:)
535:x
533:(
531:f
525:I
521:x
509:I
502:)
500:g
496:m
492:f
488:I
486:(
482:A
446:p
442:m
435:.
431:p
424:m
411:x
409:(
406:j
404:h
389:x
387:(
384:i
382:g
377:,
375:x
371:n
360:â
356:â
352:f
330:p
327:,
321:,
318:1
315:=
312:j
308:,
305:0
302:=
299:)
296:x
293:(
288:j
284:h
273:m
270:,
264:,
261:1
258:=
255:i
251:,
248:0
242:)
239:x
236:(
231:i
227:g
219:o
216:t
212:t
209:c
206:e
203:j
200:b
197:u
194:s
185:)
182:x
179:(
176:f
168:x
114:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.