Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions
Pages 1008 - 1035
Abstract
This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by “standard” inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given.
References
[1]
Arutyunov AV Optimality Conditions: Abnormal and Degenerate Problems 2000 Dordrecht Kluwer
[2]
Arutyunov AV Covering of nonlinear maps on a cone in neighborhoods of irregular points Math. Notes 2005 77 4 447-460
[3]
Arutyunov AV and Izmailov AF Directional stability theorem and directional metric regularity Math. Oper. Res. 2006 31 3 526-543
[4]
Arutyunov AV and Izmailov AF Stability of possibly nonisolated solutions of constrained equations with applications to complementarity and equilibrium problems Set-Valued Var. Anal. 2018 26 327-352
[5]
Arutyunov AV and Izmailov AF Covering on a convex set in the absence of Robinson’s regularity SIAM J. Optim. 2020 30 1 604-629
[6]
Arutyunov, A.V., Magaril-Il’yaev, G.G., Tikhomirov, V.M.: Pontryagin Maximum Principle. Proof and Applications. Factorial Press, Moscow (2006). (in Russian)
[7]
Avakov ER Extremum conditions for smooth problems with equality-type constraints USSR Comput. Math. Math. Phys. 1985 25 3 24-32
[8]
Avakov ER Theorems on estimates in the neighborhood of a singular point of a mapping Math. Notes 1990 47 5 425-432
[9]
Bonnans JF and Shapiro A Perturbation Analysis of Optimization Problems 2000 New York Springer
[10]
Cibulka R, Fabian M, and Kruger AY On semiregularity of mappings J. Math. Anal. Appl. 2019 473 2 811-836
[11]
Cibulka R, Fabian M, and Roubal T An inverse mapping theorem in Fréchet–Montel spaces Set-Valued Var. Anal. 2020 28 195-208
[12]
Cibulka R and Roubal T On ranges of non-linear operators Set-Valued Var. Anal. 2022 30 798-810
[13]
Dontchev AL and Rockafellar RT Implicit Functions and Solution Mappings 2014 2 New York Springer
[14]
Evtushenko, Y.G., Purtov, V.A.: Sufficient conditions for a minimum for nonlinear programming problems. Sov. Math. Dokl. 30(2), 313–316 (1984)
[15]
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
[16]
Fischer, A., Izmailov, A.F., Jelitte, M.: Newton-type methods near critical solutions of piecewise smooth nonlinear equations. Comput. Optim. Appl. 80, 587–615 (2021)
[17]
Halkin, H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM J. Control 12(2), 229–236 (1974)
[18]
Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979)
[19]
Izmailov AF, Kurennoy AS, and Solodov MV Critical solutions of nonlinear equations: local attraction for Newton-type methods Math. Program. 2018 167 355-379
[20]
Izmailov AF, Kurennoy AS, and Solodov MV Critical solutions of nonlinear equations: stability issues Math. Program. 2018 168 475-507
[21]
Izmailov AF and Solodov MV Error bounds for -regular mappings with Lipschitzian derivatives and their applications Math. Program. 2001 89 413-435
[22]
Izmailov AF and Solodov MV The theory of 2-regularity for mappings with Lipschitzian derivatives and its applications to optimality conditions Math. Oper. Res. 2002 27 3 614-635
[23]
Kanzow C Some equation-based methods for the nonlinear complementarity problem Optim. Methods Softw. 1994 3 4 327-340
[24]
Mordukhovich BS Variational Analysis and Applications 2018 Cham Springer
[25]
Oberlin C and Wright SJ An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problems Math. Program. 2009 117 355-386
[26]
Reif, J., Cibulka, R.: On reachable states of nonlinear dynamical systems. In: Proceedings of the 5th International Conference Aplimat, pp. 309–314. Bratislava (2006)
[27]
Robinson SM Stability theory for systems of inequalities, Part II: differentiable nonlinear systems SIAM J. Numer. Anal. 1976 13 4 497-513
[28]
Robinson, S.M.: Local structure of feasible sets in nonlinear programming, Part III: stability and sensitivity. In: Cornet, B., Nguyen, V.H., Vial, J.P., (Eds.) Nonlinear Analysis and Optimization. Mathematical Programming Study, vol. 30, pp. 45–66. Springer, Berlin (1987)
[29]
Shapiro A On concepts of directional differentiability J. Optim. Theory Appl. 1990 66 3 477-487
Recommendations
Newton-type methods near critical solutions of piecewise smooth nonlinear equations
AbstractIt is well-recognized that in the presence of singular (and in particular nonisolated) solutions of unconstrained or constrained smooth nonlinear equations, the existence of critical solutions has a crucial impact on the behavior of various Newton-...
The Levenberg–Marquardt method: an overview of modern convergence theories and more
AbstractThe Levenberg–Marquardt method is a fundamental regularization technique for the Newton method applied to nonlinear equations, possibly constrained, and possibly with singular or even nonisolated solutions. We review the literature on the subject, ...
Local Attractors of Newton-Type Methods for Constrained Equations and Complementarity Problems with Nonisolated Solutions
AbstractFor constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
© The Author(s) 2023.
Publisher
Plenum Press
United States
Publication History
Published: 19 January 2023
Accepted: 05 January 2023
Received: 19 July 2022
Author Tags
Author Tags
Qualifiers
- Research-article
Funding Sources
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 16 Dec 2024