Article
| Title: | The descent algorithms for solving symmetric Pareto eigenvalue complementarity problem (English) |
| Author: | Zou, Lu |
| Author: | Lei, Yuan |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 68 |
| Issue: | 4 |
| Year: | 2023 |
| Pages: | 441-465 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | 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. (English) |
| Keyword: | Pareto eigenvalue complementarity problem |
| Keyword: | generalized eigenvalue complementarity problem |
| Keyword: | nonlinear complementarity function |
| Keyword: | descent algorithm |
| MSC: | 65F10 |
| MSC: | 65F20 |
| MSC: | 65F22 |
| MSC: | 65K10 |
| idZBL: | Zbl 07729506 |
| idMR: | MR4612742 |
| DOI: | 10.21136/AM.2023.0020-22 |
| . | |
| Date available: | 2023-07-10T14:12:00Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151704 |
| . | |
| Reference: | [1] Adly, S., Rammal, H.: A new method for solving Pareto eigenvalue complementarity problems.Comput. Optim. Appl. 55 (2013), 703-731. Zbl 1296.90124, MR 3071170, 10.1007/s10589-013-9534-y |
| Reference: | [2] Adly, S., Seeger, A.: A nonsmooth algorithm for cone-constrained eigenvalue problems.Comput. Optim. Appl. 49 (2011), 299-318. Zbl 1220.90128, MR 2795719, 10.1007/s10589-009-9297-7 |
| Reference: | [3] Barker, G. P.: Theory of cones.Linear Algebra Appl 39 (1981), 263-291. Zbl 0467.15002, MR 0625256, 10.1016/0024-3795(81)90310-4 |
| Reference: | [4] Brás, C. P., Fischer, A., Júdice, J., Schönefeld, K., Seifert, S.: A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem.Appl. Math. Comput. 294 (2017), 36-48. Zbl 1411.90335, MR 3558259, 10.1016/j.amc.2016.09.005 |
| Reference: | [5] Chen, J.-S.: On some NCP-functions based on the generalized Fischer-Burmeister function.Asia-Pac. J. Oper. Res. 24 (2007), 401-420. Zbl 1141.90557, MR 2335554, 10.1142/S0217595907001292 |
| Reference: | [6] Chen, J.-S., Pan, S.: A family of NCP functions and a descent method for the nonlinear complementarity problem.Comput. Optim. Appl. 40 (2008), 389-404. Zbl 1153.90542, MR 2411201, 10.1007/s10589-007-9086-0 |
| Reference: | [7] Cottle, R. W., Pang, J.-S., Stone, R. E.: The Linear Complementarity Problem.Computer Science and Scientific Computing. Academic Press, Boston (1992). Zbl 0757.90078, MR 1150683, 10.1137/1.9780898719000 |
| Reference: | [8] Júdice, J. J., Raydan, M., Rosa, S. S., Santos, S. A.: On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm.Numer. Algorithms 47 (2008), 391-407. Zbl 1144.65042, MR 2393205, 10.1007/s11075-008-9194-7 |
| Reference: | [9] Júdice, J. J., Sherali, H. D., Ribeiro, I. M.: The eigenvalue complementarity problem.Comput. Optim. Appl. 37 (2007), 139-156. Zbl 1181.90261, MR 2325654, 10.1007/s10589-007-9017-0 |
| Reference: | [10] Kučera, M.: A new method for obtaining eigenvalues of variational inequalities: Operators with multiple eigenvalue.Czech. Math. J. 32 (1982), 197-207. Zbl 0621.49005, MR 0654056, 10.21136/CMJ.1982.101796 |
| Reference: | [11] Le, V. K.: Some global bifurcation results for variational inequalities.J. Differ. Equations 131 (1996), 39-78. Zbl 0863.49008, MR 1415046, 10.1006/jdeq.1996.0156 |
| Reference: | [12] Thi, H. A. Le, Moeini, M., Dinh, T. Pham, Júdice, J.: A DC programming approach for solving the symmetric eigenvalue complementarity problem.Comput. Optim. Appl. 51 (2012), 1097-1117. Zbl 1241.90153, MR 2891930, 10.1007/s10589-010-9388-5 |
| Reference: | [13] Ma, C.: The semismooth and smoothing Newton methods for solving Pareto eigenvalue problem.Appl. Math. Modelling 36 (2012), 279-287. Zbl 1236.65058, MR 2835011, 10.1016/j.apm.2011.05.045 |
| Reference: | [14] Martins, J. A. C., Costa, A. Pinto Da: Stability of finite-dimensional nonlinear elastic systems with unilateral contact and friction.Int. J. Solids Struct. 37 (2000), 2519-2564. Zbl 0959.74048, MR 1757063, 10.1016/S0020-7683(98)00291-1 |
| Reference: | [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. |
| Reference: | [16] Costa, A. Pinto Da, Martins, J. A. C.: Computation of bifurcations and instabilities in some frictional contact problems.Available at \mathbbokenlink{https://www.researchgate.net/{publication/278629570_Computation_of_bifurcations_and_instabilities_in_}{some_frictional_contact_problems}} (2001), 15 pages. |
| Reference: | [17] Costa, A. Pinto Da, Martins, J. A. C., Figueiredo, I. N., Júdice, J. J.: The directional instability problem in systems with frictional contacts.Comput. Methods Appl. Mech. Eng. 193 (2004), 357-384. Zbl 1075.74596, MR 2031232, 10.1016/j.cma.2003.09.013 |
| Reference: | [18] Costa, A. Pinto Da, Seeger, A.: Cone-constrained eigenvalue problems: Theory and algorithms.Comput. Optim. Appl. 45 (2010), 25-57. Zbl 1193.65039, MR 2594595, 10.1007/s10589-008-9167-8 |
| Reference: | [19] M. Queiroz, J. Júdice, C. Humes, Jr.: The symmetric eigenvalue complementarity problem.Math. Comput. 73 (2004), 1849-1863. Zbl 1119.90059, MR 2059739, 10.1090/S0025-5718-03-01614-4 |
| Reference: | [20] Quittner, P.: Spectral analysis of variational inequalities.Commentat. Math. Univ. Carol. 27 (1986), 605-629. MR 0873631 |
| Reference: | [21] Seeger, A.: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions.Linear Algebra Appl. 292 (1999), 1-14. Zbl 1016.90067, MR 1689301, 10.1016/S0024-3795(99)00004-X |
| Reference: | [22] Seeger, A., Torki, M.: On eigenvalues induced by a cone constraint.Linear Algebra Appl. 372 (2003), 181-206. Zbl 1046.15008, MR 1999147, 10.1016/S0024-3795(03)00553-6 |
| Reference: | [23] Seeger, A., Torki, M.: Local minima of quadratic forms on convex cones.J. Glob. Optim. 44 (2009), 1-28. Zbl 1179.90255, MR 2496064, 10.1007/s10898-007-9225-2 |
| Reference: | [24] Seeger, A., Vicente-Perez, J.: On cardinality of Pareto spectra.Electron. J. Linear Algebra 22 (2011), 758-766. Zbl 1254.15015, MR 2831028, 10.13001/1081-3810.1472 |
| Reference: | [25] Tam, B.-S.: The Perron generalized eigenspace and the spectral cone of a cone-preserving map.Linear Algebra Appl. 393 (2004), 375-429. Zbl 1062.15013, MR 2098599, 10.1016/j.laa.2004.08.020 |
| Reference: | [26] Zcghloul, T., Villechaise, B.: Stress waves in a sliding contact. Part 1: Experimental study.Tribology Series 31 (1996), 33-37. 10.1016/S0167-8922(08)70767-3 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
