| Title:
|
Solving convex program via Lagrangian decomposition (English) |
| Author:
|
Knobloch, Matthias |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
40 |
| Issue:
|
5 |
| Year:
|
2004 |
| Pages:
|
[595]-610 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider general convex large-scale optimization problems in finite dimensions. Under usual assumptions concerning the structure of the constraint functions, the considered problems are suitable for decomposition approaches. Lagrangian-dual problems are formulated and solved by applying a well-known cutting-plane method of level-type. The proposed method is capable to handle infinite function values. Therefore it is no longer necessary to demand the feasible set with respect to the non-dualized constraints to be bounded. The paper primarily deals with the description of an appropriate oracle. We first discuss the realization of the oracle under appropriate assumptions for generic convex problems. Afterwards we show that for convex quadratic programs the algorithm of the oracle is universally applicable. (English) |
| Keyword:
|
level method |
| Keyword:
|
cutting-plane methods |
| Keyword:
|
decomposition methods |
| Keyword:
|
convex programming |
| Keyword:
|
nonsmooth programming |
| MSC:
|
65K05 |
| MSC:
|
90C06 |
| MSC:
|
90C25 |
| MSC:
|
90C30 |
| idZBL:
|
Zbl 1249.90198 |
| idMR:
|
MR2120999 |
| . |
| Date available:
|
2009-09-24T20:04:10Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135619 |
| . |
| Reference:
|
[1] Auslender A.: Existence of optimal solutions and duality results under weak conditions.Math. Programming 88 (2001), 45–59 Zbl 0980.90063, MR 1765892, 10.1007/PL00011377 |
| Reference:
|
[2] Beer K., Gol’stejn E. G., Sokolov N. A.: Minimization of a nondifferentiable, convex function, defined not everywhere.Optimization 51 (2002), 6, 819–840 Zbl 1036.65050, MR 1941716, 10.1080/0233193021000066446 |
| Reference:
|
[3] Beer K., Gol’stejn E. G., Sokolov N. A.: Utilization of the Level-method for Primal Decomposition in Linear Programming Problems.Preprint 2000–13, Faculty of Mathematics, University of Technology, Chemnitz 2000 |
| Reference:
|
[4] Beer K., Knobloch M.: Utilization of the Level Method for Dual Decomposition in Convex Quadratic Programming.Preprint 2002-4, Faculty of Mathematics, University of Technology, Chemnitz 2002 |
| Reference:
|
[5] Belousov E. G.: Introduction to Convex Analysis and Integer Programming (in Russian).Izdatel’stvo Moskovskogo Universiteta, Moskau 1977 MR 0475840 |
| Reference:
|
[6] Ekeland I., Témam R.: Convex analysis and variational problems.Unabridged, corrected republication of the 1976 English original. Society for Industrial and Applied Mathematics (SIAM), Philadelphia 1999 Zbl 0322.90046, MR 1727362 |
| Reference:
|
[7] Goffin J. L., Vial J. P.: Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method.Optimization Methods and Software 17 (2002), 5, 805–867 Zbl 1065.90060, MR 1953822, 10.1080/1055678021000060829a |
| Reference:
|
[8] Kiwiel K. C.: Methods of Descent for Nondifferentiable Optimization.(Lecture Notes in Mathematics 1133), Springer Verlag, Heidelberg 1985 Zbl 0561.90059, MR 0797754 |
| Reference:
|
[9] Kiwiel K. C.: Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities.Math. Programming 69B (1995), 89–105 Zbl 0857.90101, MR 1354433, 10.1007/BF01585554 |
| Reference:
|
[10] Lémarechal C., Nemirovskii, A., Nesterov Y.: New variants of bundle methods.Math. Programming 69 (1995), 111–147 MR 1354434, 10.1007/BF01585555 |
| Reference:
|
[11] Nožička F., Guddat J., Hollatz, H., Bank B.: Theorie der linearen parametrischen Optimierung.Akademie Verlag, Berlin 1974 Zbl 0284.90053 |
| Reference:
|
[12] Mifflin R.: An algorithm for constrained optimization with semi-smooth functions.Math. Oper. Res. 2 (1997), 2, 191–207 MR 0474815, 10.1287/moor.2.2.191 |
| Reference:
|
[13] Richter K.: Lösungsverfahren für konvexe Optimierungsaufgaben mit Umrandungsstruktur auf der Grundlage gleichzeitiger primal-dualer Dekomposition.Dissertation, TU Chemnitz 2000 Zbl 1130.90300, MR 1870565 |
| Reference:
|
[14] Shor N. Z.: Minimization Methods for Non-differentiable Functions.Springer Verlag, Berlin 1985 Zbl 0561.90058, MR 0775136 |
| Reference:
|
[15] Tuy H.: Convex Analysis and Global Optimization.Kluwer, Dordrecht 1998 Zbl 0904.90156, MR 1615096 |
| Reference:
|
[16] Unger T.: Erweiterungen der Levelmethode zur Lösung konvexer Optimierungsaufgaben.Dissertation, Shaker Verlag, Aachen 2003 |
| . |
