1108:
1593:
does not use dual variables but rather removes the constraints and instead penalizes deviations from the constraint. The method is conceptually simple but usually augmented
Lagrangian methods are preferred in practice since the penalty method suffers from ill-conditioning issues.
1183:
The above inequality tells us that if we minimize the maximum value we obtain from the relaxed problem, we obtain a tighter limit on the objective value of our original problem. Thus we can address the original problem by instead exploring the partially dualized problem
898:
54:, which imposes a cost on violations. These added costs are used instead of the strict inequality constraints in the optimization. In practice, this relaxed problem can often be solved more easily than the original problem.
786:
be nonnegative weights, we get penalized if we violate the constraint (2), and we are also rewarded if we satisfy the constraint strictly. The above system is called the
Lagrangian relaxation of our original problem.
784:
1837:. Egon Balas (foreword) (Translated by Steven Vajda from the (1983 Paris: Dunod) French ed.). Chichester: A Wiley-Interscience Publication. John Wiley & Sons, Ltd. pp. xxviii+489.
1380:
656:
367:
312:
828:
147:
106:
1554:, to find feasible solutions to the original problem, then we can iterate until the best upper bound and the cost of the best feasible solution converge to a desired tolerance.
1436:
712:
562:
511:
1257:
413:
1293:
1223:
1532:
1173:
1144:
886:
857:
1645:
1584:
1463:
230:
1103:{\displaystyle c^{T}{\hat {x}}\leq c^{T}{\hat {x}}+{\tilde {\lambda }}^{T}(b_{2}-A_{2}{\hat {x}})\leq c^{T}{\bar {x}}+{\tilde {\lambda }}^{T}(b_{2}-A_{2}{\bar {x}})}
455:
189:
1552:
1505:
is a candidate upper bound to the problem, the smallest of which is kept as the best upper bound. If we additionally employ a heuristic, probably seeded by the
1503:
1483:
257:
1945:
Bragin, Mikhail A.; Luh, Peter B.; Yan, Joseph H.; Yu, Nanpeng and Stern, Gary A. (2015). "Convergence of the
Surrogate Lagrangian Relaxation Method,"
830:
values, the optimal result to the
Lagrangian relaxation problem will be no smaller than the optimal result to the original problem. To see this, let
47:
by a simpler problem. A solution to the relaxed problem is an approximate solution to the original problem, and provides useful information.
2005:
1917:
724:
1842:
1804:
1757:
1726:
1695:
1661:
1629:
1962:
Bragin, Mikhail A.; Tucker, Emily, L. (2022) "Surrogate "Level-Based" Lagrangian
Relaxation for mixed-integer linear programming,"
1789:
Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May 15–19, 2000
1682:. Universitext (Second revised ed. of translation of 1997 French ed.). Berlin: Springer-Verlag. pp. xiv+490.
2000:
1566:
is quite similar in spirit to the
Lagrangian relaxation method, but adds an extra term, and updates the dual parameters
1310:
586:
57:
The problem of maximizing the
Lagrangian function of the dual variables (the Lagrangian multipliers) is the Lagrangian
1563:
317:
262:
36:
798:
28:
44:
1752:. Grundlehren der Mathematischen Wissenschaften . Vol. 306. Berlin: Springer-Verlag. pp. xviii+346.
1721:. Grundlehren der Mathematischen Wissenschaften . Vol. 305. Berlin: Springer-Verlag. pp. xviii+417.
111:
1673:
1677:
76:
1874:(1963). "Generalized Lagrange multiplier method for solving problems of optimum allocation of resources".
1771:(reprint of the 1970 Macmillan ed.). Mineola, New York: Dover Publications, Inc. pp. xiii+523.
1398:
674:
524:
473:
1977:
1784:
1745:
1714:
1669:
40:
1236:
1830:
372:
51:
1891:
1818:
1639:
1613:
1269:
1199:
1912:
1791:. Lecture Notes in Computer Science. Vol. 2241. Berlin: Springer-Verlag. pp. 112–156.
1508:
1149:
1120:
862:
833:
1971:
1750:
Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods
1838:
1800:
1753:
1722:
1691:
1657:
1625:
1609:
1569:
1448:
206:
1953:
1926:
1883:
1792:
1683:
430:
164:
1938:
1903:
1852:
1814:
1776:
1736:
1705:
1586:
in a more principled manner. It was introduced in the 1970s and has been used extensively.
1934:
1899:
1848:
1810:
1772:
1732:
1701:
1617:
1590:
1537:
1488:
1468:
242:
1994:
1871:
1822:
58:
1984:
1911:
Kiwiel, Krzysztof C.; Larsson, Torbjörn; Lindberg, P. O. (August 2007).
17:
1445:
A Lagrangian relaxation algorithm thus proceeds to explore the range of feasible
1957:
1687:
1146:
is feasible in the original problem and the second inequality is true because
70:
888:
be the optimal solution to the
Lagrangian relaxation. We can then see that
1796:
1930:
1787:(2001). "Lagrangian relaxation". In Michael JĂĽnger and Denis Naddef (ed.).
1887:
1719:
Convex analysis and minimization algorithms, Volume I: Fundamentals
1895:
1465:
values while seeking to minimize the result returned by the inner
50:
The method penalizes violations of inequality constraints using a
779:{\displaystyle \lambda =(\lambda _{1},\ldots ,\lambda _{m_{2}})}
1985:
Toy
Examples for Plotkin-Shmoys-Tardos and Arora-Kale solvers
795:
Of particular use is the property that for any fixed set of
1299:
1188:
892:
575:
419:
153:
1679:
Numerical optimization: Theoretical and practical aspects
859:
be the optimal solution to the original problem, and let
1913:"Lagrangian relaxation via ballstep subgradient methods"
571:
We may introduce the constraint (2) into the objective:
1175:
is the optimal solution to the
Lagrangian relaxation.
1859:. Editions Tec & Doc, Paris, 2008. xxx+711 pp. ).
1668:
Bonnans, J. Frédéric; Gilbert, J. Charles;
1572:
1540:
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1272:
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1202:
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1123:
901:
865:
836:
801:
727:
677:
589:
527:
476:
433:
375:
320:
265:
245:
209:
167:
114:
79:
1179:
Iterating towards a solution of the original problem
1857:Programmation mathématique: Théorie et algorithmes
1622:Network Flows: Theory, Algorithms and Applications
1578:
1546:
1526:
1497:
1477:
1457:
1430:
1374:
1287:
1251:
1217:
1167:
1138:
1102:
880:
851:
822:
778:
706:
650:
556:
505:
449:
407:
361:
306:
251:
224:
183:
141:
100:
1375:{\displaystyle c^{T}x+\lambda ^{T}(b_{2}-A_{2}x)}
651:{\displaystyle c^{T}x+\lambda ^{T}(b_{2}-A_{2}x)}
1947:Journal of Optimization Theory and Applications.
1835:Mathematical programming: Theory and algorithms
362:{\displaystyle A_{2}\in \mathbb {R} ^{m_{2},n}}
307:{\displaystyle A_{1}\in \mathbb {R} ^{m_{1},n}}
8:
1644:: CS1 maint: multiple names: authors list (
823:{\displaystyle {\tilde {\lambda }}\succeq 0}
1571:
1539:
1513:
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1406:
1400:
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1125:
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1042:
1041:
1026:
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1019:
998:
997:
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978:
965:
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953:
938:
937:
931:
913:
912:
906:
900:
867:
866:
864:
838:
837:
835:
803:
802:
800:
765:
760:
741:
726:
698:
682:
676:
636:
623:
610:
594:
588:
548:
532:
526:
497:
481:
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438:
432:
393:
380:
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345:
340:
336:
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290:
285:
281:
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264:
244:
208:
172:
166:
127:
123:
122:
113:
92:
88:
87:
78:
1748:(1993). "14 Duality for Practitioners".
142:{\displaystyle A\in \mathbb {R} ^{m,n}}
1637:
1769:Optimization theory for large systems
1117:The first inequality is true because
101:{\displaystyle x\in \mathbb {R} ^{n}}
7:
1654:Nonlinear Programming: 2nd Edition.
1918:Mathematics of Operations Research
25:
1485:problem. Each value returned by
1431:{\displaystyle A_{1}x\leq b_{1}}
707:{\displaystyle A_{1}x\leq b_{1}}
557:{\displaystyle A_{2}x\leq b_{2}}
506:{\displaystyle A_{1}x\leq b_{1}}
1855:. (2008 Second ed., in French:
1744:Hiriart-Urruty, Jean-Baptiste;
1713:Hiriart-Urruty, Jean-Baptiste;
239:If we split the constraints in
1972:doi:10.1038/s41598-022-26264-1
1652:Bertsekas, Dimitri P. (1999).
1518:
1369:
1340:
1282:
1276:
1252:{\displaystyle \lambda \geq 0}
1212:
1206:
1159:
1130:
1097:
1091:
1059:
1047:
1031:
1009:
1003:
971:
959:
943:
918:
872:
843:
808:
773:
734:
645:
616:
1:
1978:Lagrangian Relaxation Example
1767:Lasdon, Leon S. (2002).
1674:Sagastizábal, Claudia A.
408:{\displaystyle m_{1}+m_{2}=m}
1987:in CSTheory Stack Exchange.
1564:augmented Lagrangian method
1288:{\displaystyle P(\lambda )}
1218:{\displaystyle P(\lambda )}
2022:
2006:Relaxation (approximation)
1527:{\displaystyle {\bar {x}}}
1168:{\displaystyle {\bar {x}}}
1139:{\displaystyle {\hat {x}}}
881:{\displaystyle {\bar {x}}}
852:{\displaystyle {\hat {x}}}
791:The LR solution as a bound
71:linear programming problem
1958:10.1007/s10957-014-0561-3
1688:10.1007/978-3-540-35447-5
415:we may write the system:
149:, of the following form:
29:mathematical optimization
1579:{\displaystyle \lambda }
1458:{\displaystyle \lambda }
225:{\displaystyle Ax\leq b}
65:Mathematical description
45:constrained optimization
1797:10.1007/3-540-45586-8_4
69:Suppose we are given a
43:a difficult problem of
1931:10.1287/moor.1070.0261
1580:
1548:
1528:
1499:
1479:
1459:
1432:
1376:
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1253:
1219:
1169:
1140:
1104:
882:
853:
824:
780:
708:
652:
558:
507:
451:
450:{\displaystyle c^{T}x}
409:
363:
308:
253:
226:
185:
184:{\displaystyle c^{T}x}
143:
102:
1888:10.1287/opre.11.3.399
1581:
1549:
1529:
1500:
1480:
1460:
1433:
1377:
1290:
1254:
1220:
1170:
1141:
1105:
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854:
825:
781:
709:
653:
559:
508:
452:
410:
364:
309:
254:
227:
186:
144:
103:
33:Lagrangian relaxation
1570:
1538:
1509:
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373:
318:
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243:
207:
165:
112:
77:
2001:Convex optimization
1980:, in AlgNotes Blog.
1876:Operations Research
1656:Athena Scientific.
1534:values returned by
52:Lagrange multiplier
1983:Neal Young, 2012,
1964:Scientific Reports
1785:Lemaréchal, Claude
1746:Lemaréchal, Claude
1715:Lemaréchal, Claude
1670:Lemaréchal, Claude
1614:Thomas L. Magnanti
1576:
1544:
1524:
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1215:
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878:
849:
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648:
554:
503:
447:
405:
359:
304:
249:
222:
181:
139:
98:
18:Dual decomposition
1872:Everett, Hugh III
1624:. Prentice Hall.
1610:Ravindra K. Ahuja
1547:{\displaystyle P}
1521:
1498:{\displaystyle P}
1478:{\displaystyle P}
1441:
1440:
1262:
1261:
1162:
1133:
1113:
1112:
1094:
1050:
1034:
1006:
962:
946:
921:
875:
846:
811:
717:
716:
567:
566:
252:{\displaystyle A}
235:
234:
37:relaxation method
16:(Redirected from
2013:
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1266:where we define
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27:In the field of
21:
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2012:
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1991:
1990:
1910:
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1558:Related methods
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23:
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15:
12:
11:
5:
2019:
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2009:
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1993:
1992:
1989:
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1981:
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1960:
1952:(1): 173-201,
1943:
1925:(3): 669–686.
1908:
1882:(3): 399–417.
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1618:James B. Orlin
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1591:penalty method
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24:
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10:
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3:
2:
2018:
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1844:0-471-90170-9
1840:
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1808:
1806:3-540-42877-1
1802:
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1759:3-540-56852-2
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1728:3-540-56850-6
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