79:
51:
86:. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.
1189:
585:
1693:
2032:
874:
1517:
438:
1525:
1871:
754:
297:
1184:{\displaystyle L_{i}((x_{j}^{k})_{k,j},(\lambda _{k})_{k},(\mu _{j})_{j})=f^{i}(x^{i})+\sum _{k=2}^{m}\lambda _{k}(z_{k}-f^{k}(x^{k}))+\sum _{j=1}^{n}\mu _{j}\left(b_{j}-\sum _{k=1}^{m}x_{j}^{k}\right)}
345:
430:
387:
142:
824:
1388:
2157:
Zitzler, Knowles and Thiele compare several algorithms for Pareto-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity.
1863:
1232:
661:
2104:
Since generating the entire Pareto front is often computationally-hard, there are algorithms for computing an approximate Pareto-front. For example, Legriel et al. call a set
215:
182:
1272:
1735:
1380:
1342:
862:
2090:
1787:
1304:
1813:
1755:
580:{\displaystyle P(Y)=\{y^{\prime }\in Y:\;\{y^{\prime \prime }\in Y:\;y^{\prime \prime }\succ y^{\prime },y^{\prime }\neq y^{\prime \prime }\;\}=\emptyset \}.}
1688:{\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{k}}}=-\lambda _{k}f_{x_{j}^{k}}^{i}-\mu _{j}=0{\text{ for }}k=2,\ldots ,m{\text{ and }}j=1,\ldots ,n,}
2256:
Costa, N. R., & Lourenço, J. A., "Exploring Pareto
Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds.,
2475:
Kim, I. Y.; de Weck, O. L. (2005). "Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation".
2027:{\displaystyle {\frac {f_{x_{j}^{i}}^{i}}{f_{x_{s}^{i}}^{i}}}={\frac {\mu _{j}}{\mu _{s}}}={\frac {f_{x_{j}^{k}}^{k}}{f_{x_{s}^{k}}^{k}}}.}
2592:
Mavrotas, George (2009). "Effective implementation of the ε-constraint method in Multi-Objective
Mathematical Programming problems".
2670:
Zitzler, Eckart; Knowles, Joshua; Thiele, Lothar (2008), Branke, Jürgen; Deb, Kalyanmoy; Miettinen, Kaisa; Słowiński, Roman (eds.),
666:
220:
54:
Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point
2174:
2691:
2654:
305:
392:
2518:
Marler, R. Timothy; Arora, Jasbir S. (2009). "The weighted sum method for multi-objective optimization: new insights".
2285:
1512:{\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{i}}}=f_{x_{j}^{i}}^{1}-\mu _{j}=0{\text{ for }}j=1,\ldots ,n,}
350:
106:
763:
2215:
83:
2627:
Legriel, Julien; Le
Guernic, Colas; Cotton, Scott; Maler, Oded (2010). Esparza, Javier; Majumdar, Rupak (eds.).
2309:
Tomoiagă, Bogdan; Chindriş, Mircea; Sumper, Andreas; Sudria-Andreu, Antoni; Villafafila-Robles, Roberto (2013).
596:
2561:"On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization".
2037:
Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.
1818:
20:
1274:
are the vectors of multipliers. Taking the partial derivative of the
Lagrangian with respect to each good
2311:"Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II"
1197:
610:
595:
A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the
2722:
2717:
191:
158:
2430:
Godfrey, P.; Shipley, R.; Gryz, J. (2006). "Algorithms and
Analyses for Maximal Vector Computation".
2366:
1237:
2444:
2277:
The welfare economics of public policy : a practical approach to project and policy evaluation
1701:
1347:
1309:
829:
2439:
2361:
2075:
1760:
1277:
599:
is the same for all consumers. A formal statement can be derived by considering a system with
2261:
2219:
2244:
2057:
1792:
43:. It allows the designer to restrict attention to the set of efficient choices, and to make
865:
8:
2543:
2500:
2457:
2412:
2120:
2048:
for computing the Pareto frontier of a finite set of alternatives have been studied in
1740:
78:
2687:
2678:, Lecture Notes in Computer Science, Berlin, Heidelberg: Springer, pp. 373–404,
2650:
2609:
2574:
2547:
2535:
2492:
2375:
2291:
2281:
2504:
2679:
2640:
2601:
2566:
2527:
2484:
2461:
2449:
2416:
2402:
2371:
2332:
2322:
2064:
2049:
36:
2182:
2683:
2645:
2628:
2352:
Nielsen, Frank (1996). "Output-sensitive peeling of convex and maximal layers".
2280:. Hueth, Darrell L., Schmitz, Andrew. Cheltenham, UK: E. Elgar. pp. 18–21.
2671:
2605:
2570:
2531:
2488:
2453:
302:
We assume that the preferred directions of criteria values are known. A point
103:), may be more formally described as follows. Consider a system with function
2711:
2613:
2578:
2539:
2496:
149:
2295:
2258:
Transactions on
Engineering Technologies: World Congress on Engineering 2014
47:
within this set, rather than considering the full range of every parameter.
153:
2407:
2390:
2275:
40:
2629:"Approximating the Pareto Front of Multi-criteria Optimization Problems"
2211:
74:
are not strictly dominated by any other, and hence lie on the frontier.
2337:
2327:
2310:
2236:
2045:
44:
2676:
Multiobjective
Optimization: Interactive and Evolutionary Approaches
58:
is not on the Pareto frontier because it is dominated by both point
2308:
2240:
2633:
Tools and
Algorithms for the Construction and Analysis of Systems
2204:
Introduction to
Optimization Analysis in Hydrosystem Engineering
2207:
50:
749:{\displaystyle x^{i}=(x_{1}^{i},x_{2}^{i},\ldots ,x_{n}^{i})}
292:{\displaystyle Y=\{y\in \mathbb {R} ^{m}:\;y=f(x),x\in X\;\}}
2626:
2069:"The scalarization algorithm" or the method of weighted sums
864:. To find the Pareto optimal allocation, we maximize the
1382:
gives the following system of first-order conditions:
556:
514:
491:
404:
340:{\displaystyle y^{\prime \prime }\in \mathbb {R} ^{m}}
317:
2078:
1874:
1821:
1795:
1763:
1743:
1704:
1528:
1391:
1350:
1312:
1280:
1240:
1200:
877:
832:
766:
669:
613:
441:
395:
353:
308:
223:
194:
161:
109:
2669:
2202:
Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z.,
2388:
2231:Jahan, A., Edwards, K. L., & Bahraminasab, M.,
425:{\displaystyle y^{\prime \prime }\succ y^{\prime }}
347:is preferred to (strictly dominates) another point
2563:IEEE Transactions on Systems, Man, and Cybernetics
2429:
2084:
2026:
1857:
1807:
1781:
1749:
1729:
1687:
1511:
1374:
1336:
1298:
1266:
1226:
1183:
856:
818:
748:
655:
607:goods, and a utility function of each consumer as
579:
424:
381:
339:
291:
209:
176:
136:
2672:"Quality Assessment of Pareto Set Approximations"
2389:Kung, H. T.; Luccio, F.; Preparata, F.P. (1975).
2709:
2273:
590:
2511:
382:{\displaystyle y^{\prime }\in \mathbb {R} ^{m}}
137:{\displaystyle f:X\rightarrow \mathbb {R} ^{m}}
2468:
819:{\displaystyle \sum _{i=1}^{m}x_{j}^{i}=b_{j}}
2520:Structural and Multidisciplinary Optimization
2477:Structural and Multidisciplinary Optimization
1865:. The above first-order condition imply that
2585:
1852:
1834:
571:
562:
480:
457:
286:
230:
188:is the feasible set of criterion vectors in
2517:
2391:"On finding the maxima of a set of vectors"
2474:
561:
505:
479:
432:. The Pareto frontier is thus written as:
285:
254:
2644:
2443:
2406:
2365:
2336:
2326:
369:
327:
241:
197:
164:
124:
39:solutions. The concept is widely used in
2591:
77:
49:
2639:. Berlin, Heidelberg: Springer: 69–83.
2351:
2095:Multi-objective Evolutionary Algorithms
2710:
2172:
1858:{\displaystyle j,s\in \{1,\ldots ,n\}}
16:Set of all Pareto efficient situations
2635:. Lecture Notes in Computer Science.
2260:(Berlin/Heidelberg: Springer, 2015),
2052:and power engineering. They include:
756:is the vector of goods, both for all
2063:"The maximum vector problem" or the
2594:Applied Mathematics and Computation
2139:-approximation of any Pareto front
13:
1737:denotes the partial derivative of
1547:
1532:
1410:
1395:
1227:{\displaystyle (\lambda _{k})_{k}}
656:{\displaystyle z_{i}=f^{i}(x^{i})}
568:
553:
540:
527:
511:
488:
465:
417:
401:
359:
314:
14:
2734:
2147:dimensions can be found using (1/
2099:
2233:Multi-criteria Decision Analysis
760:. The feasibility constraint is
210:{\displaystyle \mathbb {R} ^{m}}
177:{\displaystyle \mathbb {R} ^{n}}
2663:
2620:
2554:
84:production-possibility frontier
2423:
2382:
2354:Information Processing Letters
2345:
2302:
2267:
2250:
2225:
2196:
2166:
2040:
1267:{\displaystyle (\mu _{j})_{j}}
1255:
1241:
1215:
1201:
1085:
1082:
1069:
1043:
1006:
993:
977:
968:
954:
942:
928:
910:
891:
888:
743:
683:
650:
637:
451:
445:
270:
264:
119:
1:
2160:
1730:{\displaystyle f_{x_{j}^{i}}}
1375:{\displaystyle k=1,\ldots ,m}
1337:{\displaystyle j=1,\ldots ,n}
857:{\displaystyle j=1,\ldots ,n}
597:marginal rate of substitution
591:Marginal rate of substitution
152:of feasible decisions in the
90:
2684:10.1007/978-3-540-88908-3_14
2565:. SMC-1 (3): 296–297. 1971.
2376:10.1016/0020-0190(96)00116-0
21:multi-objective optimization
7:
2646:10.1007/978-3-642-12002-2_6
10:
2739:
2606:10.1016/j.amc.2009.03.037
2571:10.1109/TSMC.1971.4308298
2532:10.1007/s00158-009-0460-7
2489:10.1007/s00158-005-0557-6
2454:10.1007/s00778-006-0029-7
2274:Just, Richard E. (2004).
2085:{\displaystyle \epsilon }
1782:{\displaystyle x_{j}^{i}}
1299:{\displaystyle x_{j}^{k}}
2135:. They observe that an
1808:{\displaystyle k\neq i}
2086:
2028:
1859:
1809:
1783:
1751:
1731:
1689:
1513:
1376:
1338:
1300:
1268:
1228:
1185:
1160:
1111:
1032:
858:
820:
787:
750:
657:
581:
426:
383:
341:
293:
211:
178:
138:
87:
75:
2408:10.1145/321906.321910
2087:
2058:maxima of a point set
2029:
1860:
1810:
1784:
1752:
1732:
1690:
1514:
1377:
1339:
1301:
1269:
1229:
1186:
1140:
1091:
1012:
859:
821:
767:
751:
658:
582:
427:
384:
342:
294:
212:
179:
139:
95:The Pareto frontier,
81:
53:
2115:of the Pareto-front
2092:-constraints method"
2076:
1872:
1819:
1793:
1761:
1741:
1702:
1526:
1389:
1348:
1310:
1278:
1238:
1198:
875:
830:
764:
667:
611:
439:
393:
351:
306:
221:
192:
159:
107:
35:) is the set of all
2018:
2011:
1991:
1984:
1930:
1923:
1903:
1896:
1778:
1724:
1610:
1603:
1564:
1460:
1453:
1427:
1295:
1175:
908:
802:
742:
718:
700:
2395:Journal of the ACM
2121:Hausdorff distance
2119:, if the directed
2082:
2024:
1997:
1992:
1970:
1965:
1909:
1904:
1882:
1877:
1855:
1805:
1779:
1764:
1747:
1727:
1710:
1685:
1589:
1584:
1550:
1509:
1439:
1434:
1413:
1372:
1334:
1296:
1281:
1264:
1224:
1181:
1161:
894:
854:
816:
788:
746:
728:
704:
686:
653:
577:
422:
379:
337:
289:
207:
174:
134:
88:
76:
2723:Pareto efficiency
2718:Power engineering
2693:978-3-540-88908-3
2656:978-3-642-12002-2
2328:10.3390/en6031439
2019:
1958:
1931:
1750:{\displaystyle f}
1659:
1633:
1566:
1483:
1429:
2730:
2703:
2702:
2701:
2700:
2667:
2661:
2660:
2648:
2624:
2618:
2617:
2589:
2583:
2582:
2558:
2552:
2551:
2515:
2509:
2508:
2472:
2466:
2465:
2447:
2427:
2421:
2420:
2410:
2386:
2380:
2379:
2369:
2349:
2343:
2342:
2340:
2330:
2306:
2300:
2299:
2271:
2265:
2254:
2248:
2229:
2223:
2200:
2194:
2193:
2191:
2190:
2181:. Archived from
2170:
2091:
2089:
2088:
2083:
2050:computer science
2033:
2031:
2030:
2025:
2020:
2017:
2012:
2010:
2005:
1990:
1985:
1983:
1978:
1964:
1959:
1957:
1956:
1947:
1946:
1937:
1932:
1929:
1924:
1922:
1917:
1902:
1897:
1895:
1890:
1876:
1864:
1862:
1861:
1856:
1814:
1812:
1811:
1806:
1788:
1786:
1785:
1780:
1777:
1772:
1757:with respect to
1756:
1754:
1753:
1748:
1736:
1734:
1733:
1728:
1726:
1725:
1723:
1718:
1694:
1692:
1691:
1686:
1660:
1657:
1634:
1631:
1623:
1622:
1609:
1604:
1602:
1597:
1583:
1582:
1567:
1565:
1563:
1558:
1545:
1544:
1543:
1530:
1518:
1516:
1515:
1510:
1484:
1481:
1473:
1472:
1459:
1454:
1452:
1447:
1430:
1428:
1426:
1421:
1408:
1407:
1406:
1393:
1381:
1379:
1378:
1373:
1343:
1341:
1340:
1335:
1305:
1303:
1302:
1297:
1294:
1289:
1273:
1271:
1270:
1265:
1263:
1262:
1253:
1252:
1233:
1231:
1230:
1225:
1223:
1222:
1213:
1212:
1190:
1188:
1187:
1182:
1180:
1176:
1174:
1169:
1159:
1154:
1136:
1135:
1121:
1120:
1110:
1105:
1081:
1080:
1068:
1067:
1055:
1054:
1042:
1041:
1031:
1026:
1005:
1004:
992:
991:
976:
975:
966:
965:
950:
949:
940:
939:
924:
923:
907:
902:
887:
886:
863:
861:
860:
855:
825:
823:
822:
817:
815:
814:
801:
796:
786:
781:
755:
753:
752:
747:
741:
736:
717:
712:
699:
694:
679:
678:
662:
660:
659:
654:
649:
648:
636:
635:
623:
622:
586:
584:
583:
578:
560:
559:
544:
543:
531:
530:
518:
517:
495:
494:
469:
468:
431:
429:
428:
423:
421:
420:
408:
407:
388:
386:
385:
380:
378:
377:
372:
363:
362:
346:
344:
343:
338:
336:
335:
330:
321:
320:
298:
296:
295:
290:
250:
249:
244:
216:
214:
213:
208:
206:
205:
200:
183:
181:
180:
175:
173:
172:
167:
143:
141:
140:
135:
133:
132:
127:
37:Pareto efficient
2738:
2737:
2733:
2732:
2731:
2729:
2728:
2727:
2708:
2707:
2706:
2698:
2696:
2694:
2668:
2664:
2657:
2625:
2621:
2590:
2586:
2560:
2559:
2555:
2516:
2512:
2473:
2469:
2428:
2424:
2387:
2383:
2367:10.1.1.259.1042
2350:
2346:
2307:
2303:
2288:
2272:
2268:
2255:
2251:
2230:
2226:
2201:
2197:
2188:
2186:
2171:
2167:
2163:
2102:
2077:
2074:
2073:
2043:
2013:
2006:
2001:
1996:
1986:
1979:
1974:
1969:
1963:
1952:
1948:
1942:
1938:
1936:
1925:
1918:
1913:
1908:
1898:
1891:
1886:
1881:
1875:
1873:
1870:
1869:
1820:
1817:
1816:
1794:
1791:
1790:
1789:. Now, fix any
1773:
1768:
1762:
1759:
1758:
1742:
1739:
1738:
1719:
1714:
1709:
1705:
1703:
1700:
1699:
1658: and
1656:
1632: for
1630:
1618:
1614:
1605:
1598:
1593:
1588:
1578:
1574:
1559:
1554:
1546:
1539:
1535:
1531:
1529:
1527:
1524:
1523:
1482: for
1480:
1468:
1464:
1455:
1448:
1443:
1438:
1422:
1417:
1409:
1402:
1398:
1394:
1392:
1390:
1387:
1386:
1349:
1346:
1345:
1311:
1308:
1307:
1290:
1285:
1279:
1276:
1275:
1258:
1254:
1248:
1244:
1239:
1236:
1235:
1218:
1214:
1208:
1204:
1199:
1196:
1195:
1170:
1165:
1155:
1144:
1131:
1127:
1126:
1122:
1116:
1112:
1106:
1095:
1076:
1072:
1063:
1059:
1050:
1046:
1037:
1033:
1027:
1016:
1000:
996:
987:
983:
971:
967:
961:
957:
945:
941:
935:
931:
913:
909:
903:
898:
882:
878:
876:
873:
872:
831:
828:
827:
810:
806:
797:
792:
782:
771:
765:
762:
761:
737:
732:
713:
708:
695:
690:
674:
670:
668:
665:
664:
644:
640:
631:
627:
618:
614:
612:
609:
608:
593:
552:
548:
539:
535:
526:
522:
510:
506:
487:
483:
464:
460:
440:
437:
436:
416:
412:
400:
396:
394:
391:
390:
373:
368:
367:
358:
354:
352:
349:
348:
331:
326:
325:
313:
309:
307:
304:
303:
245:
240:
239:
222:
219:
218:
201:
196:
195:
193:
190:
189:
168:
163:
162:
160:
157:
156:
128:
123:
122:
108:
105:
104:
93:
29:Pareto frontier
17:
12:
11:
5:
2736:
2726:
2725:
2720:
2705:
2704:
2692:
2662:
2655:
2619:
2600:(2): 455–465.
2584:
2553:
2526:(6): 853–862.
2510:
2483:(2): 105–116.
2467:
2445:10.1.1.73.6344
2422:
2381:
2344:
2321:(3): 1439–55.
2301:
2286:
2266:
2249:
2224:
2195:
2179:www.cenaero.be
2175:"Pareto Front"
2164:
2162:
2159:
2113:-approximation
2101:
2100:Approximations
2098:
2097:
2096:
2093:
2081:
2070:
2067:
2061:
2042:
2039:
2035:
2034:
2023:
2016:
2009:
2004:
2000:
1995:
1989:
1982:
1977:
1973:
1968:
1962:
1955:
1951:
1945:
1941:
1935:
1928:
1921:
1916:
1912:
1907:
1901:
1894:
1889:
1885:
1880:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1804:
1801:
1798:
1776:
1771:
1767:
1746:
1722:
1717:
1713:
1708:
1696:
1695:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1629:
1626:
1621:
1617:
1613:
1608:
1601:
1596:
1592:
1587:
1581:
1577:
1573:
1570:
1562:
1557:
1553:
1549:
1542:
1538:
1534:
1520:
1519:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1479:
1476:
1471:
1467:
1463:
1458:
1451:
1446:
1442:
1437:
1433:
1425:
1420:
1416:
1412:
1405:
1401:
1397:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1293:
1288:
1284:
1261:
1257:
1251:
1247:
1243:
1221:
1217:
1211:
1207:
1203:
1192:
1191:
1179:
1173:
1168:
1164:
1158:
1153:
1150:
1147:
1143:
1139:
1134:
1130:
1125:
1119:
1115:
1109:
1104:
1101:
1098:
1094:
1090:
1087:
1084:
1079:
1075:
1071:
1066:
1062:
1058:
1053:
1049:
1045:
1040:
1036:
1030:
1025:
1022:
1019:
1015:
1011:
1008:
1003:
999:
995:
990:
986:
982:
979:
974:
970:
964:
960:
956:
953:
948:
944:
938:
934:
930:
927:
922:
919:
916:
912:
906:
901:
897:
893:
890:
885:
881:
853:
850:
847:
844:
841:
838:
835:
813:
809:
805:
800:
795:
791:
785:
780:
777:
774:
770:
745:
740:
735:
731:
727:
724:
721:
716:
711:
707:
703:
698:
693:
689:
685:
682:
677:
673:
652:
647:
643:
639:
634:
630:
626:
621:
617:
603:consumers and
592:
589:
588:
587:
576:
573:
570:
567:
564:
558:
555:
551:
547:
542:
538:
534:
529:
525:
521:
516:
513:
509:
504:
501:
498:
493:
490:
486:
482:
478:
475:
472:
467:
463:
459:
456:
453:
450:
447:
444:
419:
415:
411:
406:
403:
399:
376:
371:
366:
361:
357:
334:
329:
324:
319:
316:
312:
288:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
253:
248:
243:
238:
235:
232:
229:
226:
204:
199:
171:
166:
131:
126:
121:
118:
115:
112:
92:
89:
15:
9:
6:
4:
3:
2:
2735:
2724:
2721:
2719:
2716:
2715:
2713:
2695:
2689:
2685:
2681:
2677:
2673:
2666:
2658:
2652:
2647:
2642:
2638:
2634:
2630:
2623:
2615:
2611:
2607:
2603:
2599:
2595:
2588:
2580:
2576:
2572:
2568:
2564:
2557:
2549:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2514:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2471:
2463:
2459:
2455:
2451:
2446:
2441:
2437:
2433:
2426:
2418:
2414:
2409:
2404:
2401:(4): 469–76.
2400:
2396:
2392:
2385:
2377:
2373:
2368:
2363:
2359:
2355:
2348:
2339:
2334:
2329:
2324:
2320:
2316:
2312:
2305:
2297:
2293:
2289:
2287:1-84542-157-4
2283:
2279:
2278:
2270:
2263:
2259:
2253:
2246:
2242:
2238:
2234:
2228:
2221:
2217:
2213:
2209:
2205:
2199:
2185:on 2020-02-26
2184:
2180:
2176:
2169:
2165:
2158:
2155:
2153:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2112:
2107:
2094:
2079:
2071:
2068:
2066:
2065:skyline query
2062:
2059:
2055:
2054:
2053:
2051:
2047:
2038:
2021:
2014:
2007:
2002:
1998:
1993:
1987:
1980:
1975:
1971:
1966:
1960:
1953:
1949:
1943:
1939:
1933:
1926:
1919:
1914:
1910:
1905:
1899:
1892:
1887:
1883:
1878:
1868:
1867:
1866:
1849:
1846:
1843:
1840:
1837:
1831:
1828:
1825:
1822:
1802:
1799:
1796:
1774:
1769:
1765:
1744:
1720:
1715:
1711:
1706:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1627:
1624:
1619:
1615:
1611:
1606:
1599:
1594:
1590:
1585:
1579:
1575:
1571:
1568:
1560:
1555:
1551:
1540:
1536:
1522:
1521:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1477:
1474:
1469:
1465:
1461:
1456:
1449:
1444:
1440:
1435:
1431:
1423:
1418:
1414:
1403:
1399:
1385:
1384:
1383:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1291:
1286:
1282:
1259:
1249:
1245:
1219:
1209:
1205:
1177:
1171:
1166:
1162:
1156:
1151:
1148:
1145:
1141:
1137:
1132:
1128:
1123:
1117:
1113:
1107:
1102:
1099:
1096:
1092:
1088:
1077:
1073:
1064:
1060:
1056:
1051:
1047:
1038:
1034:
1028:
1023:
1020:
1017:
1013:
1009:
1001:
997:
988:
984:
980:
972:
962:
958:
951:
946:
936:
932:
925:
920:
917:
914:
904:
899:
895:
883:
879:
871:
870:
869:
867:
851:
848:
845:
842:
839:
836:
833:
811:
807:
803:
798:
793:
789:
783:
778:
775:
772:
768:
759:
738:
733:
729:
725:
722:
719:
714:
709:
705:
701:
696:
691:
687:
680:
675:
671:
645:
641:
632:
628:
624:
619:
615:
606:
602:
598:
574:
565:
549:
545:
536:
532:
523:
519:
507:
502:
499:
496:
484:
476:
473:
470:
461:
454:
448:
442:
435:
434:
433:
413:
409:
397:
389:, written as
374:
364:
355:
332:
322:
310:
300:
282:
279:
276:
273:
267:
261:
258:
255:
251:
246:
236:
233:
227:
224:
202:
187:
169:
155:
151:
147:
129:
116:
113:
110:
102:
98:
85:
80:
73:
69:
65:
61:
57:
52:
48:
46:
42:
38:
34:
30:
27:(also called
26:
22:
2697:, retrieved
2675:
2665:
2636:
2632:
2622:
2597:
2593:
2587:
2562:
2556:
2523:
2519:
2513:
2480:
2476:
2470:
2435:
2432:VLDB Journal
2431:
2425:
2398:
2394:
2384:
2360:(5): 255–9.
2357:
2353:
2347:
2318:
2314:
2304:
2276:
2269:
2257:
2252:
2232:
2227:
2203:
2198:
2187:. Retrieved
2183:the original
2178:
2173:proximedia.
2168:
2156:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2116:
2110:
2109:
2105:
2103:
2044:
2036:
1697:
1193:
757:
604:
600:
594:
301:
217:, such that
185:
154:metric space
145:
100:
96:
94:
71:
67:
63:
59:
55:
33:Pareto curve
32:
28:
25:Pareto front
24:
18:
2262:pp. 399–412
2235:, 2nd ed. (
2220:pp. 111–148
2131:is at most
2041:Computation
150:compact set
41:engineering
2712:Categories
2699:2021-10-08
2338:2117/18257
2212:Heidelberg
2189:2018-10-08
2161:References
2046:Algorithms
866:Lagrangian
91:Definition
62:and point
2614:0096-3003
2579:0018-9472
2548:122325484
2540:1615-147X
2497:1615-147X
2440:CiteSeerX
2362:CiteSeerX
2245:pp. 63–65
2243:, 2013),
2237:Amsterdam
2218:, 2014),
2154:queries.
2080:ϵ
1950:μ
1940:μ
1844:…
1832:∈
1800:≠
1674:…
1648:…
1616:μ
1612:−
1576:λ
1572:−
1548:∂
1533:∂
1498:…
1466:μ
1462:−
1411:∂
1396:∂
1364:…
1326:…
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1206:λ
1142:∑
1138:−
1114:μ
1093:∑
1057:−
1035:λ
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846:…
769:∑
723:…
569:∅
557:′
554:′
546:≠
541:′
528:′
520:≻
515:′
512:′
497:∈
492:′
489:′
471:∈
466:′
418:′
410:≻
405:′
402:′
365:∈
360:′
323:∈
318:′
315:′
280:∈
237:∈
120:→
66:. Points
45:tradeoffs
2505:18237050
2438:: 5–28.
2315:Energies
2296:58538348
2241:Elsevier
2216:Springer
2123:between
144:, where
2462:7374749
2417:2698043
2690:
2653:
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2364:
2294:
2284:
2208:Berlin
1698:where
1194:where
663:where
184:, and
23:, the
2544:S2CID
2501:S2CID
2458:S2CID
2413:S2CID
2072:"The
2056:"The
148:is a
2688:ISBN
2651:ISBN
2637:6015
2610:ISSN
2575:ISSN
2536:ISSN
2493:ISSN
2292:OCLC
2282:ISBN
2127:and
1815:and
1344:and
1306:for
1234:and
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70:and
2680:doi
2641:doi
2602:doi
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2567:doi
2528:doi
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2450:doi
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2372:doi
2333:hdl
2323:doi
2143:in
2108:an
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2264:.
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2210:/
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2192:.
2151:)
2149:ε
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99:(
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
