Regularity of Newton’s iteration for general parametric variational system. (English) Zbl 1426.49028
Summary: In this work, we apply the contraction mapping principle for set-valued mappings to study the Lipschitz-like property of the solution mapping to general parametric variational system obtained via canonical perturbation of a generalized equation. Then, under the assumptions of partial metric regularity and partial Lipschitz-like property, we obtained quadratic convergence of the iterative sequence when started close enough to the solution and provided estimates for the convergence parameters. Upon reconsidering all associated infinite sequences of Newton’s iterates as the image of a mapping defined on a sequence space, we established efficient conditions ensuring such a mapping to possess partial Lipschitz-like properties and also provided estimates of the moduli.
MSC:
| 49M15 | Newton-type methods |
| 49J53 | Set-valued and variational analysis |
| 49M37 | Numerical methods based on nonlinear programming |
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 90C31 | Sensitivity, stability, parametric optimization |
| 49M25 | Discrete approximations in optimal control |
Keywords:
parametric variational system; (partial) metric regularity; (partial) Lipschitz-like property; contraction mapping principle; set-valued mappingsReferences:
| [1] | Adly, S.; Cibulka, R.; Ngai, HV, Newton’s method for solving inclusions using set-valued approximations, SIAM J. Optim., 25, 159-184 (2015) · Zbl 1356.49041 · doi:10.1137/130926730 |
| [2] | Adly, S.; Ngai, HV; Nguyen, VV, Newton’s method for solving generalized equations: Kantorovich’s and Smale’s approaches, J. Math. Anal. Appl., 439, 396-418 (2016) · Zbl 1339.49014 · doi:10.1016/j.jmaa.2016.02.047 |
| [3] | Adly, S.; Ngai, HV; Nguyen, VV, Stability of metric regularity with set-valued perturbations and application to Newton’s method for solving generalized equations, Set Valued Var. Anal., 25, 543-567 (2017) · Zbl 1393.49012 · doi:10.1007/s11228-017-0438-3 |
| [4] | Aragon Artacho, FJ; Dontchev, AL; Gaydu, M.; Geoffroy, MH; Veliov, VM, Metric regularity of Newton’s iteration, SIAM J. Optim., 49, 339-362 (2011) · Zbl 1218.49024 · doi:10.1137/100792585 |
| [5] | Cibulka, R.; Dontchev, AL; Geoffroy, MH, Inexact newton methods and Dennis-Moré theorem for nonsmooth generalized equations, SIAM J. Control Optim., 53, 1003-1019 (2015) · Zbl 1321.49044 · doi:10.1137/140969476 |
| [6] | Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009) · Zbl 1178.26001 · doi:10.1007/978-0-387-87821-8 |
| [7] | Dontchev, AL; Rockafellar, RT, Newton’s method for generalized equations: a sequential implicit function theorem, Math. Progr. Ser. B, 123, 139-159 (2010) · Zbl 1190.49024 · doi:10.1007/s10107-009-0322-5 |
| [8] | Dontchev, AL; Rockafellar, RT, Convergence of inexact Newton methods for generalized equations, Math. Progr. Ser. B, 139, 115-137 (2013) · Zbl 1272.49047 · doi:10.1007/s10107-013-0664-x |
| [9] | Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90001 |
| [10] | Ioffe, AD, On regularity concepts in variational analysis, J. Fixed Point Theory Appl., 8, 339-363 (2010) · Zbl 1205.49023 · doi:10.1007/s11784-010-0021-0 |
| [11] | Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, New York (2014) · Zbl 1304.49001 · doi:10.1007/978-3-319-04247-3 |
| [12] | Lu, S.; Robinson, SM, Variational inequalities over perturbed polyhedral convex sets, Math. Oper. Res., 33, 689-711 (2008) · Zbl 1218.90192 · doi:10.1287/moor.1070.0297 |
| [13] | Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I. Basic Theory. II. Applications. Springer, Berlin (2006) · Zbl 1100.49002 · doi:10.1007/3-540-31247-1 |
| [14] | Mordukhovich, BS; Nghia, TTA, Local monotonicity and full stability for parametric variational systems, SIAM J. Optim., 26, 1032-1059 (2016) · Zbl 1337.49030 · doi:10.1137/15M1036890 |
| [15] | Mordukhovich, BS; Nghia, TTA; Pham, DT, Full stability of general parametric variational systems, Set Valued Var. Anal., 26, 911-946 (2018) · Zbl 1406.49011 · doi:10.1007/s11228-018-0474-7 |
| [16] | Ouyang, W.; Zhang, B.; He, Q., Lipschitzian stability of fully parameterized generalized equations, Appl. Anal., 97, 1717-1729 (2018) · Zbl 1396.49012 · doi:10.1080/00036811.2017.1336228 |
| [17] | Ouyang, W.; Zhang, B., Newton’s method for fully parameterized generalized equations, Optimization, 67, 2061-2080 (2018) · Zbl 1412.49058 · doi:10.1080/02331934.2018.1512605 |
| [18] | Robinson, SM, Generalized equations and their solutions. I. Basic theory. Point-to-set maps and mathematical programming, Math. Progr. Stud., 10, 128-141 (1979) · Zbl 0404.90093 · doi:10.1007/BFb0120850 |
| [19] | Robinson, SM, Variational conditions with smooth constraints. Structure and analysis, Math. Progr., 97, 245-265 (2003) · Zbl 1035.90091 · doi:10.1007/s10107-003-0443-1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
