The descent algorithms for solving symmetric Pareto eigenvalue complementarity problem

Lu Zou; Yuan Lei

Applications of Mathematics (2023)

  • Volume: 68, Issue: 4, page 441-465
  • ISSN: 0862-7940

Abstract

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For the symmetric Pareto Eigenvalue Complementarity Problem (EiCP), by reformulating it as a constrained optimization problem on a differentiable Rayleigh quotient function, we present a class of descent methods and prove their convergence. The main features include: using nonlinear complementarity functions (NCP functions) and Rayleigh quotient gradient as the descent direction, and determining the step size with exact linear search. In addition, these algorithms are further extended to solve the Generalized Eigenvalue Complementarity Problem (GEiCP) derived from unilateral friction elastic systems. Numerical experiments show the efficiency of the proposed methods compared to the projected steepest descent method with less CPU time.

How to cite

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Zou, Lu, and Lei, Yuan. "The descent algorithms for solving symmetric Pareto eigenvalue complementarity problem." Applications of Mathematics 68.4 (2023): 441-465. <http://eudml.org/doc/299570>.

@article{Zou2023,
abstract = {For the symmetric Pareto Eigenvalue Complementarity Problem (EiCP), by reformulating it as a constrained optimization problem on a differentiable Rayleigh quotient function, we present a class of descent methods and prove their convergence. The main features include: using nonlinear complementarity functions (NCP functions) and Rayleigh quotient gradient as the descent direction, and determining the step size with exact linear search. In addition, these algorithms are further extended to solve the Generalized Eigenvalue Complementarity Problem (GEiCP) derived from unilateral friction elastic systems. Numerical experiments show the efficiency of the proposed methods compared to the projected steepest descent method with less CPU time.},
author = {Zou, Lu, Lei, Yuan},
journal = {Applications of Mathematics},
keywords = {Pareto eigenvalue complementarity problem; generalized eigenvalue complementarity problem; nonlinear complementarity function; descent algorithm},
language = {eng},
number = {4},
pages = {441-465},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The descent algorithms for solving symmetric Pareto eigenvalue complementarity problem},
url = {http://eudml.org/doc/299570},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Zou, Lu
AU - Lei, Yuan
TI - The descent algorithms for solving symmetric Pareto eigenvalue complementarity problem
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 441
EP - 465
AB - For the symmetric Pareto Eigenvalue Complementarity Problem (EiCP), by reformulating it as a constrained optimization problem on a differentiable Rayleigh quotient function, we present a class of descent methods and prove their convergence. The main features include: using nonlinear complementarity functions (NCP functions) and Rayleigh quotient gradient as the descent direction, and determining the step size with exact linear search. In addition, these algorithms are further extended to solve the Generalized Eigenvalue Complementarity Problem (GEiCP) derived from unilateral friction elastic systems. Numerical experiments show the efficiency of the proposed methods compared to the projected steepest descent method with less CPU time.
LA - eng
KW - Pareto eigenvalue complementarity problem; generalized eigenvalue complementarity problem; nonlinear complementarity function; descent algorithm
UR - http://eudml.org/doc/299570
ER -

References

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  1. Adly, S., Rammal, H., 10.1007/s10589-013-9534-y, Comput. Optim. Appl. 55 (2013), 703-731. (2013) Zbl1296.90124MR3071170DOI10.1007/s10589-013-9534-y
  2. Adly, S., Seeger, A., 10.1007/s10589-009-9297-7, Comput. Optim. Appl. 49 (2011), 299-318. (2011) Zbl1220.90128MR2795719DOI10.1007/s10589-009-9297-7
  3. Barker, G. P., 10.1016/0024-3795(81)90310-4, Linear Algebra Appl 39 (1981), 263-291. (1981) Zbl0467.15002MR0625256DOI10.1016/0024-3795(81)90310-4
  4. Brás, C. P., Fischer, A., Júdice, J., Schönefeld, K., Seifert, S., 10.1016/j.amc.2016.09.005, Appl. Math. Comput. 294 (2017), 36-48. (2017) Zbl1411.90335MR3558259DOI10.1016/j.amc.2016.09.005
  5. Chen, J.-S., 10.1142/S0217595907001292, Asia-Pac. J. Oper. Res. 24 (2007), 401-420. (2007) Zbl1141.90557MR2335554DOI10.1142/S0217595907001292
  6. Chen, J.-S., Pan, S., 10.1007/s10589-007-9086-0, Comput. Optim. Appl. 40 (2008), 389-404. (2008) Zbl1153.90542MR2411201DOI10.1007/s10589-007-9086-0
  7. Cottle, R. W., Pang, J.-S., Stone, R. E., 10.1137/1.9780898719000, Computer Science and Scientific Computing. Academic Press, Boston (1992). (1992) Zbl0757.90078MR1150683DOI10.1137/1.9780898719000
  8. Júdice, J. J., Raydan, M., Rosa, S. S., Santos, S. A., 10.1007/s11075-008-9194-7, Numer. Algorithms 47 (2008), 391-407. (2008) Zbl1144.65042MR2393205DOI10.1007/s11075-008-9194-7
  9. Júdice, J. J., Sherali, H. D., Ribeiro, I. M., 10.1007/s10589-007-9017-0, Comput. Optim. Appl. 37 (2007), 139-156. (2007) Zbl1181.90261MR2325654DOI10.1007/s10589-007-9017-0
  10. Kučera, M., 10.21136/CMJ.1982.101796, Czech. Math. J. 32 (1982), 197-207. (1982) Zbl0621.49005MR0654056DOI10.21136/CMJ.1982.101796
  11. Le, V. K., 10.1006/jdeq.1996.0156, J. Differ. Equations 131 (1996), 39-78. (1996) Zbl0863.49008MR1415046DOI10.1006/jdeq.1996.0156
  12. Thi, H. A. Le, Moeini, M., Dinh, T. Pham, Júdice, J., 10.1007/s10589-010-9388-5, Comput. Optim. Appl. 51 (2012), 1097-1117. (2012) Zbl1241.90153MR2891930DOI10.1007/s10589-010-9388-5
  13. Ma, C., 10.1016/j.apm.2011.05.045, Appl. Math. Modelling 36 (2012), 279-287. (2012) Zbl1236.65058MR2835011DOI10.1016/j.apm.2011.05.045
  14. Martins, J. A. C., Costa, A. Pinto Da, 10.1016/S0020-7683(98)00291-1, Int. J. Solids Struct. 37 (2000), 2519-2564. (2000) Zbl0959.74048MR1757063DOI10.1016/S0020-7683(98)00291-1
  15. Martins, J. A. C., Costa, A. Pinto Da, Bifurcations and instabilities in frictional contact problems: Theoretical relations, computational methods and numerical results, European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004 University of Jyväskylä, Jyväskylä (2004), 102095. (2004) 
  16. Costa, A. Pinto Da, Martins, J. A. C., Computation of bifurcations and instabilities in some frictional contact problems, Available at https://www.researchgate.net/publication/278629570Computationofbifurcationsandinstabilitiesinsomefrictionalcontactproblems (2001), 15 pages. (2001) 
  17. Costa, A. Pinto Da, Martins, J. A. C., Figueiredo, I. N., Júdice, J. J., 10.1016/j.cma.2003.09.013, Comput. Methods Appl. Mech. Eng. 193 (2004), 357-384. (2004) Zbl1075.74596MR2031232DOI10.1016/j.cma.2003.09.013
  18. Costa, A. Pinto Da, Seeger, A., 10.1007/s10589-008-9167-8, Comput. Optim. Appl. 45 (2010), 25-57. (2010) Zbl1193.65039MR2594595DOI10.1007/s10589-008-9167-8
  19. M. Queiroz, J. Júdice, C. Humes, Jr., 10.1090/S0025-5718-03-01614-4, Math. Comput. 73 (2004), 1849-1863. (2004) Zbl1119.90059MR2059739DOI10.1090/S0025-5718-03-01614-4
  20. Quittner, P., Spectral analysis of variational inequalities, Commentat. Math. Univ. Carol. 27 (1986), 605-629. (1986) MR0873631
  21. Seeger, A., 10.1016/S0024-3795(99)00004-X, Linear Algebra Appl. 292 (1999), 1-14. (1999) Zbl1016.90067MR1689301DOI10.1016/S0024-3795(99)00004-X
  22. Seeger, A., Torki, M., 10.1016/S0024-3795(03)00553-6, Linear Algebra Appl. 372 (2003), 181-206. (2003) Zbl1046.15008MR1999147DOI10.1016/S0024-3795(03)00553-6
  23. Seeger, A., Torki, M., 10.1007/s10898-007-9225-2, J. Glob. Optim. 44 (2009), 1-28. (2009) Zbl1179.90255MR2496064DOI10.1007/s10898-007-9225-2
  24. Seeger, A., Vicente-Perez, J., 10.13001/1081-3810.1472, Electron. J. Linear Algebra 22 (2011), 758-766. (2011) Zbl1254.15015MR2831028DOI10.13001/1081-3810.1472
  25. Tam, B.-S., 10.1016/j.laa.2004.08.020, Linear Algebra Appl. 393 (2004), 375-429. (2004) Zbl1062.15013MR2098599DOI10.1016/j.laa.2004.08.020
  26. Zcghloul, T., Villechaise, B., 10.1016/S0167-8922(08)70767-3, Tribology Series 31 (1996), 33-37. (1996) DOI10.1016/S0167-8922(08)70767-3

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