Smoothing functions and algorithm for nonsymmetric circular cone complementarity problems

Jingyong Tang; Yuefen Chen

Applications of Mathematics (2022)

  • Volume: 67, Issue: 2, page 209-231
  • ISSN: 0862-7940

Abstract

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There has been much interest in studying symmetric cone complementarity problems. In this paper, we study the circular cone complementarity problem (denoted by CCCP) which is a type of nonsymmetric cone complementarity problem. We first construct two smoothing functions for the CCCP and show that they are all coercive and strong semismooth. Then we propose a smoothing algorithm to solve the CCCP. The proposed algorithm generates an infinite sequence such that the value of the merit function converges to zero. Moreover, we show that the iteration sequence must be bounded if the solution set of the CCCP is nonempty and bounded. At last, we prove that the proposed algorithm has local superlinear or quadratical convergence under some assumptions which are much weaker than Jacobian nonsingularity assumption. Some numerical results are reported which demonstrate that our algorithm is very effective for solving CCCPs.

How to cite

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Tang, Jingyong, and Chen, Yuefen. "Smoothing functions and algorithm for nonsymmetric circular cone complementarity problems." Applications of Mathematics 67.2 (2022): 209-231. <http://eudml.org/doc/297727>.

@article{Tang2022,
abstract = {There has been much interest in studying symmetric cone complementarity problems. In this paper, we study the circular cone complementarity problem (denoted by CCCP) which is a type of nonsymmetric cone complementarity problem. We first construct two smoothing functions for the CCCP and show that they are all coercive and strong semismooth. Then we propose a smoothing algorithm to solve the CCCP. The proposed algorithm generates an infinite sequence such that the value of the merit function converges to zero. Moreover, we show that the iteration sequence must be bounded if the solution set of the CCCP is nonempty and bounded. At last, we prove that the proposed algorithm has local superlinear or quadratical convergence under some assumptions which are much weaker than Jacobian nonsingularity assumption. Some numerical results are reported which demonstrate that our algorithm is very effective for solving CCCPs.},
author = {Tang, Jingyong, Chen, Yuefen},
journal = {Applications of Mathematics},
keywords = {circular cone complementarity problem; smoothing function; smoothing algorithm; superlinear/quadratical convergence},
language = {eng},
number = {2},
pages = {209-231},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Smoothing functions and algorithm for nonsymmetric circular cone complementarity problems},
url = {http://eudml.org/doc/297727},
volume = {67},
year = {2022},
}

TY - JOUR
AU - Tang, Jingyong
AU - Chen, Yuefen
TI - Smoothing functions and algorithm for nonsymmetric circular cone complementarity problems
JO - Applications of Mathematics
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 209
EP - 231
AB - There has been much interest in studying symmetric cone complementarity problems. In this paper, we study the circular cone complementarity problem (denoted by CCCP) which is a type of nonsymmetric cone complementarity problem. We first construct two smoothing functions for the CCCP and show that they are all coercive and strong semismooth. Then we propose a smoothing algorithm to solve the CCCP. The proposed algorithm generates an infinite sequence such that the value of the merit function converges to zero. Moreover, we show that the iteration sequence must be bounded if the solution set of the CCCP is nonempty and bounded. At last, we prove that the proposed algorithm has local superlinear or quadratical convergence under some assumptions which are much weaker than Jacobian nonsingularity assumption. Some numerical results are reported which demonstrate that our algorithm is very effective for solving CCCPs.
LA - eng
KW - circular cone complementarity problem; smoothing function; smoothing algorithm; superlinear/quadratical convergence
UR - http://eudml.org/doc/297727
ER -

References

top
  1. Alizadeh, F., Goldfarb, D., 10.1007/s10107-002-0339-5, Math. Program. 95 (2003), 3-51. (2003) Zbl1153.90522MR1971381DOI10.1007/s10107-002-0339-5
  2. Alzalg, B., 10.7153/oam-11-01, Oper. Matrices 11 (2017), 1-21. (2017) Zbl1404.17046MR3602626DOI10.7153/oam-11-01
  3. Bai, Y., Gao, X., Wang, G., 10.3934/naco.2015.5.211, Numer. Algebra Control Optim. 5 (2015), 211-231. (2015) Zbl1317.90193MR3365253DOI10.3934/naco.2015.5.211
  4. Bai, Y., Ma, P., Zhang, J., 10.3934/jimo.2016.12.739, J. Ind. Manag. Optim. 12 (2016), 739-756. (2016) Zbl1327.90192MR3413849DOI10.3934/jimo.2016.12.739
  5. Burke, J., Xu, S., 10.1007/s101079900111, Math. Program. 87 (2000), 113-130. (2000) Zbl1081.90603MR1734661DOI10.1007/s101079900111
  6. Chen, J.-S., Pan, S., A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs, Pac. J. Optim. 8 (2012), 33-74. (2012) Zbl1286.90148MR2919670
  7. Chen, J.-S., Sun, D., Sun, J., 10.1016/j.orl.2007.08.005, Oper. Res. Lett. 36 (2008), 385-392. (2008) Zbl1152.90621MR2424468DOI10.1016/j.orl.2007.08.005
  8. Chen, L., Ma, C., 10.1016/j.camwa.2011.01.009, Comput. Math. Appl. 61 (2011), 1407-1418. (2011) Zbl1217.65127MR2773413DOI10.1016/j.camwa.2011.01.009
  9. Chen, X. D., Sun, D., Sun, J., 10.1023/A:1022996819381, Comput. Optim. Appl. 25 (2003), 39-56. (2003) Zbl1038.90084MR1996662DOI10.1023/A:1022996819381
  10. Chi, X., Liu, S., 10.1080/02331930701763421, Optimization 58 (2009), 965-979. (2009) Zbl1177.90318MR2572781DOI10.1080/02331930701763421
  11. Chi, X., Liu, S., 10.1016/j.cam.2007.12.023, J. Comput. Appl. Math. 223 (2009), 114-123. (2009) Zbl1155.65045MR2463105DOI10.1016/j.cam.2007.12.023
  12. Chi, X., Tao, J., Zhu, Z., Duan, F., A regularized inexact smoothing Newton method for circular cone complementarity problem, Pac. J. Optim. 13 (2017), 197-218. (2017) Zbl1386.65155MR3711669
  13. Chi, X., Wan, Z., Zhu, Z., Yuan, L., 10.1080/02331934.2016.1217861, Optimization 65 (2016), 2227-2250. (2016) Zbl1351.90149MR3564914DOI10.1080/02331934.2016.1217861
  14. Chi, X., Wei, H., Wan, Z., Zhu, Z., 10.1016/S0252-9602(17)30072-3, Acta Math. Sci., Ser. B, Engl. Ed. 37 (2017), 1262-1280. (2017) Zbl1399.90207MR3683894DOI10.1016/S0252-9602(17)30072-3
  15. Facchinei, F., Kanzow, C., 10.1137/S0363012997322935, SIAM J. Control Optim. 37 (1999), 1150-1161. (1999) Zbl0997.90085MR1691935DOI10.1137/S0363012997322935
  16. Fang, L., Feng, Z., 10.1590/S1807-03022011000300005, Comput. Appl. Math. 30 (2011), 569-588. (2011) Zbl1401.90152MR2863924DOI10.1590/S1807-03022011000300005
  17. Fukushima, M., Luo, Z.-Q., Tseng, P., 10.1137/S1052623400380365, SIAM J. Optim. 12 (2001), 436-460. (2001) Zbl0995.90094MR1885570DOI10.1137/S1052623400380365
  18. Hayashi, S., Yamashita, N., Fukushima, M., 10.1137/S1052623403421516, SIAM J. Optim. 15 (2005), 593-615. (2005) Zbl1114.90139MR2144183DOI10.1137/S1052623403421516
  19. Huang, Z.-H., Ni, T., 10.1007/s10589-008-9180-y, Comput. Optim. Appl. 45 (2010), 557-579. (2010) Zbl1198.90373MR2600896DOI10.1007/s10589-008-9180-y
  20. Jin, P., Ling, C., Shen, H., 10.1007/s10589-014-9698-0, Comput. Optim. Appl. 60 (2015), 675-695. (2015) Zbl1318.49062MR3320940DOI10.1007/s10589-014-9698-0
  21. Ke, Y.-F., Ma, C.-F., Zhang, H., 10.1007/s40314-018-0687-2, Comput. Appl. Math. 37 (2018), 6795-6820. (2018) Zbl1413.90287MR3885844DOI10.1007/s40314-018-0687-2
  22. Kheirfam, B., Wang, G., 10.3934/naco.2017011, Numer. Algebra Control Optim. 7 (2017), 171-184. (2017) Zbl1365.90271MR3665011DOI10.3934/naco.2017011
  23. Liu, W., Wang, C., 10.1007/s40314-013-0013-y, Comput. Appl. Math. 32 (2013), 89-105. (2013) Zbl1291.90272MR3101279DOI10.1007/s40314-013-0013-y
  24. Lobo, M. S., Vandenberghe, L., Boyd, S., Lebret, H., 10.1016/S0024-3795(98)10032-0, Linear Algebra Appl. 284 (1998), 193-228. (1998) Zbl0946.90050MR1655138DOI10.1016/S0024-3795(98)10032-0
  25. Ma, P., Bai, Y., Chen, J.-S., A self-concordant interior point algorithm for nonsymmetric circular cone programming, J. Nonlinear Convex Anal. 17 (2016), 225-241. (2016) Zbl1354.90095MR3472994
  26. Miao, X.-H., Guo, S., Qi, N., Chen, J.-S., 10.1007/s10589-015-9781-1, Comput. Optim. Appl. 63 (2016), 495-522. (2016) Zbl1360.90250MR3457449DOI10.1007/s10589-015-9781-1
  27. Miao, X.-H., Yang, J., Hu, S., 10.1016/j.amc.2015.07.064, Appl. Math. Comput. 269 (2015), 155-168. (2015) Zbl1410.65124MR3396768DOI10.1016/j.amc.2015.07.064
  28. Palais, R. S., Terng, C.-L., 10.1007/BFb0087442, Lecture Notes in Mathematics 1353. Springer, Berlin (1988). (1988) Zbl0658.49001MR0972503DOI10.1007/BFb0087442
  29. Pirhaji, M., Zangiabadi, M., Mansouri, H., 10.1007/s13160-018-0317-9, Japan J. Ind. Appl. Math. 35 (2018), 1103-1121. (2018) Zbl06990734MR3868787DOI10.1007/s13160-018-0317-9
  30. Qi, L., Sun, J., 10.1007/BF01581275, Math. Program. 58 (1993), 353-367. (1993) Zbl0780.90090MR1216791DOI10.1007/BF01581275
  31. Qi, L., Sun, D., Zhou, G., 10.1007/s101079900127, Math. Program. 87 (2000), 1-35. (2000) Zbl0989.90124MR1734657DOI10.1007/s101079900127
  32. Sun, D., Sun, J., 10.1007/s10107-005-0577-4, Math. Program. 103 (2005), 575-581. (2005) Zbl1099.90062MR2166550DOI10.1007/s10107-005-0577-4
  33. Tang, J., Dong, L., Zhou, J., Sun, L., 10.1080/02331934.2014.906595, Optimization 64 (2015), 1935-1955. (2015) Zbl1337.90071MR3361160DOI10.1080/02331934.2014.906595
  34. Tang, J., He, G., Dong, L., Fang, L., Zhou, J., 10.1007/s10492-013-0011-9, Appl. Math., Praha 58 (2013), 223-247. (2013) Zbl1274.90268MR3034823DOI10.1007/s10492-013-0011-9
  35. Yoshise, A., 10.1137/04061427X, SIAM J. Optim. 17 (2006), 1129-1153. (2006) Zbl1136.90039MR2274506DOI10.1137/04061427X
  36. Zhang, J., Chen, J., A smoothing Levenberg-Marquardt type method for LCP, J. Comput. Math. 22 (2004), 735-752. (2004) Zbl1068.65084MR2080440
  37. Zhou, J., Chen, J.-S., Properties of circular cone and spectral factorization associated with circular cone, J. Nonlinear Convex Anal. 14 (2013), 807-816. (2013) Zbl1294.49007MR3131148
  38. Zhou, J., Chen, J.-S., Hung, H.-F., 10.1186/1029-242X-2013-571, J. Inequal. Appl. 2013 (2013), Article ID 571, 17 pages. (2013) Zbl1297.26025MR3212998DOI10.1186/1029-242X-2013-571
  39. Zhou, J., Chen, J.-S., Mordukhovich, B. S., 10.1080/02331934.2014.951043, Optimization 64 (2015), 113-147. (2015) Zbl1338.90396MR3293544DOI10.1080/02331934.2014.951043
  40. Zhou, J., Tang, J., Chen, J.-S., 10.1007/s10957-016-0935-9, J. Optim. Theory Appl. 172 (2017), 802-823. (2017) Zbl1362.90345MR3610222DOI10.1007/s10957-016-0935-9

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