An SQP method for mathematical programs with complementarity constraints with strong convergence properties

Matus Benko; Helmut Gfrerer

Kybernetika (2016)

  • Volume: 52, Issue: 2, page 169-208
  • ISSN: 0023-5954

Abstract

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We propose an SQP algorithm for mathematical programs with complementarity constraints which solves at each iteration a quadratic program with linear complementarity constraints. We demonstrate how strongly M-stationary solutions of this quadratic program can be obtained by an active set method without using enumeration techniques. We show that all limit points of the sequence of iterates generated by our SQP method are at least M-stationary.

How to cite

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Benko, Matus, and Gfrerer, Helmut. "An SQP method for mathematical programs with complementarity constraints with strong convergence properties." Kybernetika 52.2 (2016): 169-208. <http://eudml.org/doc/281549>.

@article{Benko2016,
abstract = {We propose an SQP algorithm for mathematical programs with complementarity constraints which solves at each iteration a quadratic program with linear complementarity constraints. We demonstrate how strongly M-stationary solutions of this quadratic program can be obtained by an active set method without using enumeration techniques. We show that all limit points of the sequence of iterates generated by our SQP method are at least M-stationary.},
author = {Benko, Matus, Gfrerer, Helmut},
journal = {Kybernetika},
keywords = {SQP method; active set method; mathematical program with complementarity constraints; strong M-stationarity; SQP method; active set method; mathematical program with complementarity constraints; strong M-stationarity},
language = {eng},
number = {2},
pages = {169-208},
publisher = {Institute of Information Theory and Automation AS CR},
title = {An SQP method for mathematical programs with complementarity constraints with strong convergence properties},
url = {http://eudml.org/doc/281549},
volume = {52},
year = {2016},
}

TY - JOUR
AU - Benko, Matus
AU - Gfrerer, Helmut
TI - An SQP method for mathematical programs with complementarity constraints with strong convergence properties
JO - Kybernetika
PY - 2016
PB - Institute of Information Theory and Automation AS CR
VL - 52
IS - 2
SP - 169
EP - 208
AB - We propose an SQP algorithm for mathematical programs with complementarity constraints which solves at each iteration a quadratic program with linear complementarity constraints. We demonstrate how strongly M-stationary solutions of this quadratic program can be obtained by an active set method without using enumeration techniques. We show that all limit points of the sequence of iterates generated by our SQP method are at least M-stationary.
LA - eng
KW - SQP method; active set method; mathematical program with complementarity constraints; strong M-stationarity; SQP method; active set method; mathematical program with complementarity constraints; strong M-stationarity
UR - http://eudml.org/doc/281549
ER -

References

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  1. Anitescu, M., 10.1137/040606855, SIAM J. Optim. 16 (2005), 120-145. Zbl1099.65050MR2177772DOI10.1137/040606855
  2. Anitescu, M., 10.1137/s1052623402401221, SIAM J. Optim. 15 (2005), 1203-1236. Zbl1097.90050MR2178496DOI10.1137/s1052623402401221
  3. Biegler, L. T., Raghunathan, A. U., 10.1137/s1052623403429081, SIAM J. Optim. 15 (2005), 720-750. Zbl1077.90079MR2142858DOI10.1137/s1052623403429081
  4. DeMiguel, V., Friedlander, M. P., Nogales, F. J., Scholtes, S., 10.1137/04060754x, SIAM J. Optim. 16 (2005), 587-609. Zbl1122.90060MR2197997DOI10.1137/04060754x
  5. Facchinei, F., Jiang, H., Qi, L., 10.1007/s101070050048, Math. Programming 85 (1999), 107-134. Zbl0959.65079MR1689366DOI10.1007/s101070050048
  6. Fletcher, R., Practical Methods of Optimization 2: Constrained Optimization., John Wiley and Sons, Chichester 1981. MR0633058
  7. Fletcher, R., Leyffer, S., Solving mathematical programs with complementary constraints as nonlinear programs. Zbl1074.90044
  8. Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S., 10.1137/s1052623402407382, SIAM J. Optim. 17 (2006), 259-286. Zbl1112.90098MR2219153DOI10.1137/s1052623402407382
  9. Fukushima, M., Tseng, P., 10.1137/s1052623499363232, SIAM J. Optim. 12 (2002), 724-739. Zbl1127.65034MR1884914DOI10.1137/s1052623499363232
  10. Gfrerer, H., 10.1137/130914449, SIAM J. Optim. 24 (2014), 898-931. Zbl1298.49021MR3217222DOI10.1137/130914449
  11. Giallombardo, G., Ralph, D., 10.1007/s10107-006-0020-5, Math. Programming 112 (2008), 335-369. Zbl1145.90073MR2361928DOI10.1007/s10107-006-0020-5
  12. Hu, X. M., Ralph, D., 10.1007/s10957-004-5154-0, J. Optim. Theory Appl. 123 (2004), 365-390. MR2101411DOI10.1007/s10957-004-5154-0
  13. Izmailov, A. F., Pogosyan, A. L., Solodov, M. V., 10.1007/s10589-010-9341-7, Computational Optim. Appl. 51 (2012), 199-221. Zbl1245.90124MR2872496DOI10.1007/s10589-010-9341-7
  14. Jiang, H., Ralph, D., 10.1023/A:1008696504163, Comput. Optim. Appl. 13 (1999), 25-59. MR1704113DOI10.1023/A:1008696504163
  15. Jiang, H., Ralph, D., 10.1023/A:1022945316191, Comput. Optim. Appl. 25 (2002), 123-150. Zbl1038.90100MR1996665DOI10.1023/A:1022945316191
  16. Kadrani, A., Dussault, J. P., Benchakroun, A., 10.1137/070705490, SIAM J. Optim. 20 (2009), 78-103. Zbl1187.65064MR2496894DOI10.1137/070705490
  17. Kanzow, C., Schwartz, A., 10.1137/100802487, SIAM J. Optim. 23 (2013), 770-798. Zbl1282.65069MR3045664DOI10.1137/100802487
  18. Kanzow, C., Schwartz, A., 10.1287/moor.2014.0667, Math. Oper. Res. 40 (2015), 2, 253-275. MR3320430DOI10.1287/moor.2014.0667
  19. Leyffer, S.,  DOI
  20. Leyffer, S., López-Calva, G., Nocedal, J., 10.1137/040621065, SIAM J. Optim. 17 (2006), 52-77. Zbl1112.90095MR2219144DOI10.1137/040621065
  21. Lin, G. H., Fukushima, M., 10.1007/s10479-004-5024-z, Ann. Oper. Res. 133 (2005), 63-84. Zbl1119.90058MR2119313DOI10.1007/s10479-004-5024-z
  22. Luo, Z. Q., Pang, J. S., Ralph, D., 10.1017/cbo9780511983658, Cambridge University Press, Cambridge, New York, Melbourne 1996. Zbl1139.90003MR1419501DOI10.1017/cbo9780511983658
  23. Luo, Z. Q., Pang, J. S., Ralph, D., 10.1007/978-1-4613-0307-7_9, In: Multilevel Optimization: Algorithms, Complexity, and Applications (A. Migdalas, P. Pardalos, and P. Värbrand, eds.), Kluwer Academic Publishers, Dordrecht 1998, pp. 209-229. Zbl0897.90184MR1605239DOI10.1007/978-1-4613-0307-7_9
  24. Luo, Z. Q., Pang, J. S., Ralph, D., Wu, S. Q., 10.1007/bf02592205, Math. Programming 75 (1996), 19-76. Zbl0870.90092MR1415093DOI10.1007/bf02592205
  25. Outrata, J. V., Kočvara, M., Zowe, J., 10.1007/978-1-4757-2825-5, Kluwer Academic Publishers, Dordrecht 1998. MR1641213DOI10.1007/978-1-4757-2825-5
  26. Powell, M. J. D., 10.1007/bfb0067703, In: Numerical Analysis Dundee 1977 (G. A. Watson, ed.), Lecture Notes in Mathematics 630, Springer, Berlin, 1978, pp. 144-157. Zbl0374.65032MR0483447DOI10.1007/bfb0067703
  27. Ralph, D., Wright, S. J., 10.1080/10556780410001709439, Optim. Methods Software 19 (2004), 527-556. Zbl1097.90054MR2095351DOI10.1080/10556780410001709439
  28. Scheel, H., Scholtes, S., 10.1287/moor.25.1.1.15213, Math. Oper. Res. 25 (2000), 1-22. Zbl1073.90557MR1854317DOI10.1287/moor.25.1.1.15213
  29. Scholtes, S., 10.1137/s1052623499361233, SIAM J. Optim. 11 (2001), 918-936. Zbl1010.90086MR1855214DOI10.1137/s1052623499361233
  30. Scholtes, S., Stöhr, M., 10.1137/s0363012996306121, SIAM J. Control Optim. 37 (1999), 617-652. Zbl0922.90128MR1670641DOI10.1137/s0363012996306121
  31. Steffensen, S., Ulbrich, M., 10.1137/090748883, SIAM J. Optim. 20 (2010), 2504-2539. MR2678403DOI10.1137/090748883
  32. Stein, O., 10.1007/s10107-010-0345-y, Math. Programming 131 (2012), 71-94. Zbl1250.90094MR2886141DOI10.1007/s10107-010-0345-y
  33. Stöhr, M., Nonsmooth Trust Region Methods and their Applications to Mathematical Programs with Equilibrium Constraints., Shaker-Verlag, Aachen 1999. 
  34. Zhang, J., Liu, G., 10.1023/A:1011226232107, J. Glob. Optim. 19 (2001), 335-361. Zbl1049.90125MR1824769DOI10.1023/A:1011226232107

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