| Title:
|
A new regular multiplier embedding (English) |
| Author:
|
Bouza Allende, Gemayqzel |
| Author:
|
Guddat, Jürgen |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
236-257 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Embedding approaches can be used for solving non linear programs P. The idea is to define a one-parametric problem such that for some value of the parameter the corresponding problem is equivalent to P. A particular case is the multipliers embedding, where the solutions of the corresponding parametric problem can be interpreted as the points computed by the multipliers method on P. However, in the known cases, either path-following methods can not be applied or the necessary conditions for its convergence are fulfilled under very restrictive hypothesis. In this paper, we present a new multipliers embedding such that the objective function and the constraints of $P(t)$ are $C^3$ differentiable functions. We prove that the parametric problem satisfies the JJT-regularity generically, a necessary condition for the success of the path-following method. (English) |
| Keyword:
|
Jongen–Jonker–Twilt regularity |
| Keyword:
|
multipliers method |
| Keyword:
|
embedding |
| MSC:
|
49M30 |
| MSC:
|
90C31 |
| idZBL:
|
Zbl 1266.90178 |
| idMR:
|
MR3085395 |
| . |
| Date available:
|
2013-07-22T08:46:25Z |
| Last updated:
|
2016-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143366 |
| . |
| Reference:
|
[1] Afonso, M., Bioucas-Dias, J., Figueiredo, M.: Fast image recovery using variable splitting and constrained optimization..IEEE Trans. Signal Process. 19 (2010), 2345-2356. MR 2798930 |
| Reference:
|
[2] Andreani, R., Birgin, E. G., Martínez, J. M., Schuverdt, M. L.: On augmented Lagrangian methods with general lower-level constraints..SIAM J. Optim. 28 (2008), 1286-1309. Zbl 1151.49027, MR 2373302, 10.1137/060654797 |
| Reference:
|
[3] Avelino, C, Vicente, L. N.: Updating the multipliers associated with inequality constraints in an augmented Lagrangian multiplier method..J. Optim. Theory Appl. 199 (2003), 215-233. Zbl 1094.90045, MR 2028992, 10.1023/B:JOTA.0000005444.50285.4d |
| Reference:
|
[4] Bazaraa, M. S., Sherali, H. D., Shetty, C. M.: Non Linear Programming Theory and Algorithms..John Willey and Sons, 1993. MR 2218478 |
| Reference:
|
[5] Bertsekas, D. P.: Constrained Optimization and Lagrange Multiplier Methods..Academic Press, New York 1982. Zbl 0662.90044, MR 0690767 |
| Reference:
|
[6] Birgin, E. G., Martínez, J. M.: Augmented lagrangian method with nonmonotone penalty parameters for constrained..Optimization Online, http://www.optimization-online.org/DB_FILE/2010/06/2662.pdf 2010. Zbl 1244.90216 |
| Reference:
|
[7] Bouza, G.: A new embedding for the augmented Lagrangean method..Investigación Oper. 22 (2001), 145-153. MR 1868635 |
| Reference:
|
[8] Bouza, G., Guddat, J.: A note on embeddings for the augmented Lagrange method..Yugosl. J. Oper. Res. 20 (2010), 183-196. MR 2771069, 10.2298/YJOR1002183B |
| Reference:
|
[9] Dentcheva, D., Gollmer, R., Guddat, J., Rückmann, J.: Pathfollowing methods in non linear optimization, multipliers embedding..ZOR 41 (1995), 127-152. MR 1336625 |
| Reference:
|
[10] Dostal, Z., Friedlander, A., Santos, A.: Augmented Lagrangians with adaptative precision control for quadratic programming problems with equality constrains..Comput. Optim. Appl. 14 (1999), 37-53. MR 1704945, 10.1023/A:1008700911674 |
| Reference:
|
[11] Gollmer, R., Kausmann, U., Nowack, D., Wendler, K., Estrada, J. Bacallao: Computerprogramm PAFO..Humboldt-Universitaet, Institut fuer Mathematik 2004. |
| Reference:
|
[12] Gómez, W.: On generic quadratic penalty embeddings for non linear optimization problems..Optimization 50 (2001), 279-295. MR 1890006, 10.1080/02331930108844564 |
| Reference:
|
[13] Gómez, W., Guddat, J., Jongen, H. Th., Rückmann, J. J., Solano, C.: Curvas criticas y saltos en la optimizacion no lineal..http://www.emis.de/monographs/curvas/index.html 2000. |
| Reference:
|
[14] Guddat, J., Guerra, F., Jongen, H. Th.: Parametric Optimization: Singularities, Pathfollowing and Jumps..Teubner and John Wiley, Chichester 1990. MR 1085483 |
| Reference:
|
[15] Hirsch, M.: Differential Topology..Springer Verlag, New York 1976. Zbl 0804.57001, MR 0448362 |
| Reference:
|
[16] Iusem, A. N.: Augmented Lagrangean methods and proximal point methods for convex optimization..Investigación Oper. 8 (1999), 11-49. |
| Reference:
|
[17] Jongen, H. Th., Jonker, P., Twilt, F.: Critical sets in parametric optimization..Math. Programming 34 (1986), 333-353. Zbl 0599.90114, MR 0839608, 10.1007/BF01582234 |
| Reference:
|
[18] Jongen, H. Th., Jonker, P., Twilt, F.: On one-parametrer families of optimization problems: Equality constrains..J. Optim. Theory Appl. 48 (1986), 141-161. MR 0825389 |
| Reference:
|
[19] Li, D., Sun, X. L.: Local convexification of the Lagrangian function in non-convex poptimization..J. Optim. Theory Appl. 104 (2000), 109-120. MR 1741392, 10.1023/A:1004628822745 |
| Reference:
|
[20] Li, Z., Ierapetritou, M. G.: Production planning and scheduling integration through augmented lagrangian optimization..Comput. and Chemical Engrg. 34 (2010), 996-1006. 10.1016/j.compchemeng.2009.11.016 |
| Reference:
|
[21] Luenberger, D. G., Ye, Yinyu: Linear and Nonlinear Programming. Third edition..Internat. Ser. Oper. Res. Management Sci. Springer, New York 2008. MR 2423726 |
| Reference:
|
[22] Schmidt, R.: Eine modifizierte standard Einbettung zur Behandlung von Gleichungs und Ungleichungs Restriktionen..Master's Thesis, Humboldt Universitaet zu Berlin 2000. |
| . |
