1003:
718:
2011:
1249:
998:{\displaystyle {\begin{aligned}F({\boldsymbol {x}}+t{\boldsymbol {\delta _{sd}}})&\approx {\frac {1}{2}}\left\|{\boldsymbol {f}}({\boldsymbol {x}})+t{\boldsymbol {J}}({\boldsymbol {x}}){\boldsymbol {\delta _{sd}}}\right\|^{2}\\&=F({\boldsymbol {x}})+t{\boldsymbol {\delta _{sd}}}^{\top }{\boldsymbol {J}}^{\top }{\boldsymbol {f}}({\boldsymbol {x}})+{\frac {1}{2}}t^{2}\left\|{\boldsymbol {J}}{\boldsymbol {\delta _{sd}}}\right\|^{2}.\end{aligned}}}
75:
1058:
236:
1244:{\displaystyle t=-{\frac {{\boldsymbol {\delta _{sd}}}^{\top }{\boldsymbol {J}}^{\top }{\boldsymbol {f}}({\boldsymbol {x}})}{\left\|{\boldsymbol {J}}{\boldsymbol {\delta _{sd}}}\right\|^{2}}}={\frac {\left\|{\boldsymbol {\delta _{sd}}}\right\|^{2}}{\left\|{\boldsymbol {J}}{\boldsymbol {\delta _{sd}}}\right\|^{2}}}.}
553:
54:
along the steepest descent direction, known as Cauchy point. If the Cauchy point is outside of the trust region, it is truncated to the boundary of the latter and it is taken as the new solution. If the Cauchy point is inside the trust region, the new solution is taken at the intersection between the
1579:
707:
1447:
92:
348:
455:
627:
414:
1383:
1635:
1490:
289:
723:
50:. At each iteration, if the step from the Gauss–Newton algorithm is within the trust region, it is used to update the current solution. If not, the algorithm searches for the minimum of the
1497:
1337:
1302:
1908:
443:
643:
231:{\displaystyle F({\boldsymbol {x}})={\frac {1}{2}}\left\|{\boldsymbol {f}}({\boldsymbol {x}})\right\|^{2}={\frac {1}{2}}\sum _{i=1}^{m}\left(f_{i}({\boldsymbol {x}})\right)^{2}}
1390:
1779:
294:
548:{\displaystyle {\boldsymbol {\delta _{gn}}}=-\left({\boldsymbol {J}}^{\top }{\boldsymbol {J}}\right)^{-1}{\boldsymbol {J}}^{\top }{\boldsymbol {f}}({\boldsymbol {x}})}
1903:
1273:
561:
1599:
1050:
1026:
2504:
1917:
2397:
1681:
Lourakis, M.L.A.; Argyros, A.A. (2005). "Is
Levenberg-Marquardt the most efficient optimization algorithm for implementing bundle adjustment?".
356:
1772:
1342:
2478:
1940:
1992:
1853:
1960:
35:
1698:
1604:
1452:
2071:
1765:
1725:
Powell, M.J.D. (1970). "A new algorithm for unconstrained optimization". In Rosen, J.B.; Mangasarian, O.L.; Ritter, K. (eds.).
630:
2348:
2010:
244:
2456:
2076:
2392:
2360:
2441:
2066:
1574:{\displaystyle t{\boldsymbol {\delta _{sd}}}+s\left({\boldsymbol {\delta _{gn}}}-t{\boldsymbol {\delta _{sd}}}\right)}
2387:
2343:
1945:
446:
39:
58:
The name of the method derives from the resemblance between the construction of the dog leg step and the shape of a
2236:
2126:
1310:
1788:
1278:
23:
2311:
27:
2173:
1637:, if the Gauss–Newton step is outside the trust region but the steepest descent step is inside (dog leg step).
2355:
2254:
1970:
702:{\displaystyle {\boldsymbol {\delta _{sd}}}=-{\boldsymbol {J}}^{\top }{\boldsymbol {f}}({\boldsymbol {x}}).}
1848:
419:
2499:
2446:
2431:
2321:
2199:
1825:
1792:
1442:{\displaystyle {\frac {\Delta }{\left\|{\boldsymbol {\delta _{sd}}}\right\|}}{\boldsymbol {\delta _{sd}}}}
2335:
2301:
2204:
2146:
2027:
1833:
1813:
1750:
55:
trust region boundary and the line joining the Cauchy point and the Gauss-Newton step (dog leg step).
2382:
2209:
2121:
31:
1757:
2451:
2316:
2269:
2259:
2111:
2099:
1912:
1895:
1800:
343:{\displaystyle {\boldsymbol {x}}^{*}=\operatorname {argmin} _{\boldsymbol {x}}F({\boldsymbol {x}})}
2186:
2155:
2141:
2131:
1922:
1704:
51:
1838:
2194:
1872:
1694:
1258:
2274:
2264:
2168:
2045:
1950:
1932:
1885:
1796:
1686:
634:
43:
20:
1734:
Powell, M.J.D. (1970). "A hybrid method for nonlinear equations". In
Robinowitz, P. (ed.).
622:{\displaystyle {\boldsymbol {J}}=\left({\frac {\partial {f_{i}}}{\partial {x_{j}}}}\right)}
2290:
2278:
2163:
2050:
1984:
1955:
1584:
1035:
1011:
1449:
if both the Gauss–Newton and the steepest descent steps are outside the trust region (
2493:
2436:
2420:
83:
1708:
2374:
1880:
47:
2461:
1843:
59:
1716:
Yuan, Ya-xiang (2000). "A review of trust region algorithms for optimization".
1029:
74:
1690:
712:
The objective function is linearised along the steepest descent direction
1863:
1683:
Tenth IEEE International
Conference on Computer Vision (ICCV'05) Volume 1
409:{\displaystyle {\boldsymbol {x}}_{k}={\boldsymbol {x}}_{k-1}+\delta _{k}}
351:
2183:
1378:{\displaystyle \left\|{\boldsymbol {\delta _{gn}}}\right\|\leq \Delta }
73:
63:
2418:
2234:
2097:
2025:
1811:
1761:
1630:{\displaystyle \left\|{\boldsymbol {\delta }}\right\|=\Delta }
2009:
1485:{\displaystyle t\left\|{\boldsymbol {\delta _{sd}}}\right\|}
1339:, if the Gauss–Newton step is within the trust region (
284:{\displaystyle f_{i}:\mathbb {R} ^{n}\to \mathbb {R} }
1738:. London: Gordon and Breach Science. pp. 87–144.
1607:
1587:
1500:
1455:
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1061:
1038:
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721:
646:
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247:
95:
2373:
2334:
2300:
2289:
2247:
2182:
2154:
2140:
2110:
2059:
2038:
1983:
1931:
1894:
1871:
1862:
1824:
1736:
Numerical
Methods for Nonlinear Algebraic Equations
1629:
1593:
1573:
1484:
1441:
1377:
1331:
1296:
1275:, Powell's dog leg method selects the update step
1267:
1243:
1044:
1020:
997:
701:
621:
547:
437:
408:
342:
291:, Powell's dog leg method finds the optimal point
283:
230:
1773:
8:
1652:
1650:
1332:{\displaystyle {\boldsymbol {\delta _{gn}}}}
19:, also called Powell's hybrid method, is an
1729:. New York: Academic Press. pp. 31–66.
1297:{\displaystyle {\boldsymbol {\delta _{k}}}}
2415:
2331:
2297:
2244:
2231:
2151:
2107:
2094:
2035:
2022:
1868:
1821:
1808:
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1758:
1612:
1606:
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1556:
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1530:
1509:
1504:
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1468:
1463:
1454:
1429:
1424:
1408:
1403:
1394:
1392:
1355:
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1344:
1319:
1314:
1312:
1287:
1282:
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1260:
1230:
1215:
1210:
1205:
1193:
1179:
1174:
1167:
1156:
1141:
1136:
1131:
1116:
1108:
1102:
1097:
1090:
1080:
1075:
1071:
1060:
1037:
1013:
982:
967:
962:
957:
945:
931:
920:
912:
906:
901:
894:
884:
879:
864:
842:
827:
822:
814:
806:
792:
784:
768:
748:
743:
732:
722:
720:
688:
680:
674:
669:
652:
647:
645:
605:
600:
588:
583:
577:
565:
563:
537:
529:
523:
518:
508:
498:
492:
487:
464:
459:
457:
429:
424:
421:
400:
381:
376:
366:
361:
358:
332:
317:
304:
299:
296:
277:
276:
267:
263:
262:
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246:
222:
209:
200:
184:
173:
159:
150:
137:
129:
113:
102:
94:
2014:Optimization computes maxima and minima.
1646:
1613:
1560:
1557:
1553:
1539:
1536:
1532:
1513:
1510:
1506:
1472:
1469:
1465:
1433:
1430:
1426:
1412:
1409:
1405:
1359:
1356:
1352:
1323:
1320:
1316:
1288:
1284:
1219:
1216:
1212:
1206:
1183:
1180:
1176:
1145:
1142:
1138:
1132:
1117:
1109:
1098:
1084:
1081:
1077:
1052:is imposed to be equal to zero, giving
1032:of the last expression with respect to
971:
968:
964:
958:
921:
913:
902:
888:
885:
881:
865:
831:
828:
824:
815:
807:
793:
785:
752:
749:
745:
733:
689:
681:
670:
656:
653:
649:
566:
538:
530:
519:
499:
488:
468:
465:
461:
425:
377:
362:
333:
318:
300:
210:
138:
130:
103:
1664:
1662:
1008:To compute the value of the parameter
2210:Principal pivoting algorithm of Lemke
438:{\displaystyle {\boldsymbol {x}}^{*}}
7:
2505:Optimization algorithms and methods
1854:Successive parabolic interpolation
1624:
1396:
1372:
1262:
1103:
1091:
907:
895:
675:
597:
580:
524:
493:
14:
2174:Projective algorithm of Karmarkar
2169:Ellipsoid algorithm of Khachiyan
2072:Sequential quadratic programming
1909:Broyden–Fletcher–Goldfarb–Shanno
78:Construction of the dog leg step
30:problems, introduced in 1970 by
1255:Given a trust region of radius
2127:Reduced gradient (Frank–Wolfe)
1617:
1609:
1478:
1460:
1418:
1400:
1365:
1347:
1226:
1201:
1189:
1171:
1152:
1127:
1121:
1113:
978:
953:
925:
917:
869:
861:
838:
819:
811:
797:
789:
780:
758:
729:
693:
685:
542:
534:
337:
329:
273:
214:
206:
146:
142:
134:
125:
107:
99:
26:algorithm for the solution of
1:
2457:Spiral optimization algorithm
2077:Successive linear programming
1751:"Equation Solving Algorithms"
36:Levenberg–Marquardt algorithm
2195:Simplex algorithm of Dantzig
2067:Augmented Lagrangian methods
445:. At a given iteration, the
2521:
46:, but it uses an explicit
2474:
2427:
2414:
2398:Push–relabel maximum flow
2243:
2230:
2200:Revised simplex algorithm
2106:
2093:
2034:
2021:
2007:
1820:
1807:
1028:at the Cauchy point, the
1923:Symmetric rank-one (SR1)
1904:Berndt–Hall–Hall–Hausman
28:non-linear least squares
2447:Parallel metaheuristics
2255:Approximation algorithm
1966:Powell's dog leg method
1918:Davidon–Fletcher–Powell
1814:Unconstrained nonlinear
1268:{\displaystyle \Delta }
17:Powell's dog leg method
2432:Evolutionary algorithm
2015:
1685:. pp. 1526–1531.
1631:
1595:
1575:
1486:
1443:
1379:
1333:
1298:
1269:
1245:
1046:
1022:
999:
703:
637:direction is given by
623:
549:
439:
410:
344:
285:
232:
189:
79:
40:Gauss–Newton algorithm
2205:Criss-cross algorithm
2028:Constrained nonlinear
2013:
1834:Golden-section search
1727:Nonlinear Programming
1691:10.1109/ICCV.2005.128
1632:
1596:
1576:
1487:
1444:
1380:
1334:
1299:
1270:
1246:
1047:
1023:
1000:
704:
624:
550:
440:
411:
345:
286:
233:
169:
77:
2122:Cutting-plane method
1605:
1585:
1498:
1453:
1391:
1343:
1311:
1279:
1259:
1059:
1036:
1012:
719:
644:
562:
456:
420:
357:
295:
245:
93:
86:problem in the form
32:Michael J. D. Powell
2452:Simulated annealing
2270:Integer programming
2260:Dynamic programming
2100:Convex optimization
1961:Levenberg–Marquardt
34:. Similarly to the
2132:Subgradient method
2016:
1941:Conjugate gradient
1849:Nelder–Mead method
1627:
1591:
1571:
1482:
1439:
1375:
1329:
1294:
1265:
1241:
1042:
1018:
995:
993:
699:
619:
545:
435:
416:that converges to
406:
350:by constructing a
340:
281:
228:
80:
52:objective function
38:, it combines the
2487:
2486:
2470:
2469:
2410:
2409:
2406:
2405:
2369:
2368:
2330:
2329:
2226:
2225:
2222:
2221:
2218:
2217:
2089:
2088:
2085:
2084:
2005:
2004:
2001:
2000:
1979:
1978:
1594:{\displaystyle s}
1422:
1236:
1162:
1045:{\displaystyle t}
1021:{\displaystyle t}
939:
776:
613:
449:step is given by
167:
121:
2512:
2416:
2332:
2298:
2275:Branch and bound
2265:Greedy algorithm
2245:
2232:
2152:
2108:
2095:
2036:
2023:
1971:Truncated Newton
1886:Wolfe conditions
1869:
1822:
1809:
1782:
1775:
1768:
1759:
1754:
1739:
1730:
1721:
1712:
1669:
1666:
1657:
1654:
1636:
1634:
1633:
1628:
1620:
1616:
1600:
1598:
1597:
1592:
1580:
1578:
1577:
1572:
1570:
1566:
1565:
1564:
1563:
1544:
1543:
1542:
1518:
1517:
1516:
1491:
1489:
1488:
1483:
1481:
1477:
1476:
1475:
1448:
1446:
1445:
1440:
1438:
1437:
1436:
1423:
1421:
1417:
1416:
1415:
1395:
1384:
1382:
1381:
1376:
1368:
1364:
1363:
1362:
1338:
1336:
1335:
1330:
1328:
1327:
1326:
1303:
1301:
1300:
1295:
1293:
1292:
1291:
1274:
1272:
1271:
1266:
1250:
1248:
1247:
1242:
1237:
1235:
1234:
1229:
1225:
1224:
1223:
1222:
1209:
1198:
1197:
1192:
1188:
1187:
1186:
1168:
1163:
1161:
1160:
1155:
1151:
1150:
1149:
1148:
1135:
1124:
1120:
1112:
1107:
1106:
1101:
1095:
1094:
1089:
1088:
1087:
1072:
1051:
1049:
1048:
1043:
1027:
1025:
1024:
1019:
1004:
1002:
1001:
996:
994:
987:
986:
981:
977:
976:
975:
974:
961:
950:
949:
940:
932:
924:
916:
911:
910:
905:
899:
898:
893:
892:
891:
868:
851:
847:
846:
841:
837:
836:
835:
834:
818:
810:
796:
788:
777:
769:
757:
756:
755:
736:
708:
706:
705:
700:
692:
684:
679:
678:
673:
661:
660:
659:
635:steepest descent
628:
626:
625:
620:
618:
614:
612:
611:
610:
609:
595:
594:
593:
592:
578:
569:
554:
552:
551:
546:
541:
533:
528:
527:
522:
516:
515:
507:
503:
502:
497:
496:
491:
473:
472:
471:
444:
442:
441:
436:
434:
433:
428:
415:
413:
412:
407:
405:
404:
392:
391:
380:
371:
370:
365:
349:
347:
346:
341:
336:
322:
321:
309:
308:
303:
290:
288:
287:
282:
280:
272:
271:
266:
257:
256:
237:
235:
234:
229:
227:
226:
221:
217:
213:
205:
204:
188:
183:
168:
160:
155:
154:
149:
145:
141:
133:
122:
114:
106:
44:gradient descent
2520:
2519:
2515:
2514:
2513:
2511:
2510:
2509:
2490:
2489:
2488:
2483:
2466:
2423:
2402:
2365:
2326:
2303:
2292:
2285:
2239:
2214:
2178:
2145:
2136:
2113:
2102:
2081:
2055:
2051:Penalty methods
2046:Barrier methods
2030:
2017:
1997:
1993:Newton's method
1975:
1927:
1890:
1858:
1839:Powell's method
1816:
1803:
1786:
1749:
1746:
1733:
1724:
1720:. Vol. 99.
1715:
1701:
1680:
1677:
1672:
1667:
1660:
1655:
1648:
1644:
1608:
1603:
1602:
1583:
1582:
1552:
1531:
1529:
1525:
1505:
1496:
1495:
1464:
1459:
1451:
1450:
1425:
1404:
1399:
1389:
1388:
1351:
1346:
1341:
1340:
1315:
1309:
1308:
1283:
1277:
1276:
1257:
1256:
1254:
1211:
1204:
1200:
1199:
1175:
1170:
1169:
1137:
1130:
1126:
1125:
1096:
1076:
1074:
1073:
1057:
1056:
1034:
1033:
1010:
1009:
992:
991:
963:
956:
952:
951:
941:
900:
880:
878:
849:
848:
823:
783:
779:
778:
761:
744:
717:
716:
668:
648:
642:
641:
631:Jacobian matrix
601:
596:
584:
579:
573:
560:
559:
517:
486:
485:
481:
480:
460:
454:
453:
423:
418:
417:
396:
375:
360:
355:
354:
313:
298:
293:
292:
261:
248:
243:
242:
196:
195:
191:
190:
128:
124:
123:
91:
90:
72:
12:
11:
5:
2518:
2516:
2508:
2507:
2502:
2492:
2491:
2485:
2484:
2482:
2481:
2475:
2472:
2471:
2468:
2467:
2465:
2464:
2459:
2454:
2449:
2444:
2439:
2434:
2428:
2425:
2424:
2421:Metaheuristics
2419:
2412:
2411:
2408:
2407:
2404:
2403:
2401:
2400:
2395:
2393:Ford–Fulkerson
2390:
2385:
2379:
2377:
2371:
2370:
2367:
2366:
2364:
2363:
2361:Floyd–Warshall
2358:
2353:
2352:
2351:
2340:
2338:
2328:
2327:
2325:
2324:
2319:
2314:
2308:
2306:
2295:
2287:
2286:
2284:
2283:
2282:
2281:
2267:
2262:
2257:
2251:
2249:
2241:
2240:
2235:
2228:
2227:
2224:
2223:
2220:
2219:
2216:
2215:
2213:
2212:
2207:
2202:
2197:
2191:
2189:
2180:
2179:
2177:
2176:
2171:
2166:
2164:Affine scaling
2160:
2158:
2156:Interior point
2149:
2138:
2137:
2135:
2134:
2129:
2124:
2118:
2116:
2104:
2103:
2098:
2091:
2090:
2087:
2086:
2083:
2082:
2080:
2079:
2074:
2069:
2063:
2061:
2060:Differentiable
2057:
2056:
2054:
2053:
2048:
2042:
2040:
2032:
2031:
2026:
2019:
2018:
2008:
2006:
2003:
2002:
1999:
1998:
1996:
1995:
1989:
1987:
1981:
1980:
1977:
1976:
1974:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1937:
1935:
1929:
1928:
1926:
1925:
1920:
1915:
1906:
1900:
1898:
1892:
1891:
1889:
1888:
1883:
1877:
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84:least squares
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2442:Local search
2388:Edmonds–Karp
2344:Bellman–Ford
2114:minimization
1965:
1946:Gauss–Newton
1896:Quasi–Newton
1881:Trust region
1789:Optimization
1753:. MathWorks.
1735:
1726:
1717:
1682:
1253:
1007:
711:
633:, while the
557:
447:Gauss–Newton
240:
81:
57:
48:trust region
24:optimisation
16:
15:
2462:Tabu search
1873:Convergence
1844:Line search
1668:Yuan (2000)
70:Formulation
60:dogleg hole
2494:Categories
2293:algorithms
1801:heuristics
1793:Algorithms
1642:References
1601:such that
1030:derivative
2248:Paradigms
2147:quadratic
1864:Gradients
1826:Functions
1625:Δ
1614:δ
1554:δ
1546:−
1533:δ
1507:δ
1466:δ
1427:δ
1406:δ
1397:Δ
1373:Δ
1370:≤
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1317:δ
1285:δ
1263:Δ
1213:δ
1177:δ
1139:δ
1104:⊤
1092:⊤
1078:δ
1069:−
965:δ
908:⊤
896:⊤
882:δ
825:δ
766:≈
746:δ
676:⊤
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650:δ
598:∂
581:∂
525:⊤
510:−
494:⊤
478:−
462:δ
431:∗
398:δ
386:−
324:
306:∗
274:→
171:∑
21:iterative
2479:Software
2356:Dijkstra
2187:exchange
1985:Hessians
1951:Gradient
1709:16542484
1618:‖
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979:‖
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352:sequence
147:‖
126:‖
82:Given a
2322:Kruskal
2312:BorĹŻvka
2302:Minimum
2039:General
1797:methods
1675:Sources
629:is the
2184:Basis-
2142:Linear
2112:Convex
1956:Mirror
1913:L-BFGS
1799:, and
1707:
1697:
558:where
315:argmin
2383:Dinic
2291:Graph
1718:Iciam
1705:S2CID
1581:with
241:with
42:with
2349:SPFA
2317:Prim
1911:and
1695:ISBN
64:golf
2279:cut
2144:and
1687:doi
62:in
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