A new regular multiplier embedding

Gemayqzel Bouza Allende; Jürgen Guddat

Kybernetika (2013)

  • Volume: 49, Issue: 2, page 236-257
  • ISSN: 0023-5954

Abstract

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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.

How to cite

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Bouza Allende, Gemayqzel, and Guddat, Jürgen. "A new regular multiplier embedding." Kybernetika 49.2 (2013): 236-257. <http://eudml.org/doc/260721>.

@article{BouzaAllende2013,
abstract = {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.},
author = {Bouza Allende, Gemayqzel, Guddat, Jürgen},
journal = {Kybernetika},
keywords = {Jongen–Jonker–Twilt regularity; multipliers method; embedding; Jongen-Jonker-Twilt regularity; multipliers method; embedding},
language = {eng},
number = {2},
pages = {236-257},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A new regular multiplier embedding},
url = {http://eudml.org/doc/260721},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Bouza Allende, Gemayqzel
AU - Guddat, Jürgen
TI - A new regular multiplier embedding
JO - Kybernetika
PY - 2013
PB - Institute of Information Theory and Automation AS CR
VL - 49
IS - 2
SP - 236
EP - 257
AB - 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.
LA - eng
KW - Jongen–Jonker–Twilt regularity; multipliers method; embedding; Jongen-Jonker-Twilt regularity; multipliers method; embedding
UR - http://eudml.org/doc/260721
ER -

References

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  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. MR2798930
  2. Andreani, R., Birgin, E. G., Martínez, J. M., Schuverdt, M. L., 10.1137/060654797, SIAM J. Optim. 28 (2008), 1286-1309. Zbl1151.49027MR2373302DOI10.1137/060654797
  3. Avelino, C, Vicente, L. N., 10.1023/B:JOTA.0000005444.50285.4d, J. Optim. Theory Appl. 199 (2003), 215-233. Zbl1094.90045MR2028992DOI10.1023/B:JOTA.0000005444.50285.4d
  4. Bazaraa, M. S., Sherali, H. D., Shetty, C. M., Non Linear Programming Theory and Algorithms., John Willey and Sons, 1993. MR2218478
  5. Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods., Academic Press, New York 1982. Zbl0662.90044MR0690767
  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. Zbl1244.90216
  7. Bouza, G., A new embedding for the augmented Lagrangean method., Investigación Oper. 22 (2001), 145-153. MR1868635
  8. Bouza, G., Guddat, J., 10.2298/YJOR1002183B, Yugosl. J. Oper. Res. 20 (2010), 183-196. MR2771069DOI10.2298/YJOR1002183B
  9. Dentcheva, D., Gollmer, R., Guddat, J., Rückmann, J., Pathfollowing methods in non linear optimization, multipliers embedding., ZOR 41 (1995), 127-152. MR1336625
  10. Dostal, Z., Friedlander, A., Santos, A., 10.1023/A:1008700911674, Comput. Optim. Appl. 14 (1999), 37-53. MR1704945DOI10.1023/A:1008700911674
  11. Gollmer, R., Kausmann, U., Nowack, D., Wendler, K., Estrada, J. Bacallao, Computerprogramm PAFO., Humboldt-Universitaet, Institut fuer Mathematik 2004. 
  12. Gómez, W., 10.1080/02331930108844564, Optimization 50 (2001), 279-295. MR1890006DOI10.1080/02331930108844564
  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. 
  14. Guddat, J., Guerra, F., Jongen, H. Th., Parametric Optimization: Singularities, Pathfollowing and Jumps., Teubner and John Wiley, Chichester 1990. MR1085483
  15. Hirsch, M., Differential Topology., Springer Verlag, New York 1976. Zbl0804.57001MR0448362
  16. Iusem, A. N., Augmented Lagrangean methods and proximal point methods for convex optimization., Investigación Oper. 8 (1999), 11-49. 
  17. Jongen, H. Th., Jonker, P., Twilt, F., 10.1007/BF01582234, Math. Programming 34 (1986), 333-353. Zbl0599.90114MR0839608DOI10.1007/BF01582234
  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. MR0825389
  19. Li, D., Sun, X. L., 10.1023/A:1004628822745, J. Optim. Theory Appl. 104 (2000), 109-120. MR1741392DOI10.1023/A:1004628822745
  20. Li, Z., Ierapetritou, M. G., 10.1016/j.compchemeng.2009.11.016, Comput. and Chemical Engrg. 34 (2010), 996-1006. DOI10.1016/j.compchemeng.2009.11.016
  21. Luenberger, D. G., Ye, Yinyu, Linear and Nonlinear Programming. Third edition., Internat. Ser. Oper. Res. Management Sci. Springer, New York 2008. MR2423726
  22. Schmidt, R., Eine modifizierte standard Einbettung zur Behandlung von Gleichungs und Ungleichungs Restriktionen., Master's Thesis, Humboldt Universitaet zu Berlin 2000. 

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