Geodesic
Tilt stability in nonlinear programming under Mangasarian-Fromovitz constraint qualification
Kybernetika, Tome 49 (2013) no. 3, pp. 446-464 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The paper concerns the study of tilt stability of local minimizers in standard problems of nonlinear programming. This notion plays an important role in both theoretical and numerical aspects of optimization and has drawn a lot of attention in optimization theory and its applications, especially in recent years. Under the classical Mangasarian-Fromovitz Constraint Qualification, we establish relationships between tilt stability and some other stability notions in constrained optimization. Involving further the well-known Constant Rank Constraint Qualification, we derive new necessary and sufficient conditions for tilt-stable local minimizers.
The paper concerns the study of tilt stability of local minimizers in standard problems of nonlinear programming. This notion plays an important role in both theoretical and numerical aspects of optimization and has drawn a lot of attention in optimization theory and its applications, especially in recent years. Under the classical Mangasarian-Fromovitz Constraint Qualification, we establish relationships between tilt stability and some other stability notions in constrained optimization. Involving further the well-known Constant Rank Constraint Qualification, we derive new necessary and sufficient conditions for tilt-stable local minimizers.
Classification : 49J52, 90C30, 90C31
Keywords: variational analysis; second-order theory; generalized differentiation; tilt stability 
@article{KYB_2013_49_3_a5,
     author = {Mordukhovich, Boris S. and Outrata, Ji\v{r}{\'\i} V.},
     title = {Tilt stability in nonlinear programming under {Mangasarian-Fromovitz} constraint qualification},
     journal = {Kybernetika},
     pages = {446--464},
     year = {2013},
     volume = {49},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a5/}
}
TY  - JOUR
AU  - Mordukhovich, Boris S.
AU  - Outrata, Jiří V.
TI  - Tilt stability in nonlinear programming under Mangasarian-Fromovitz constraint qualification
JO  - Kybernetika
PY  - 2013
SP  - 446
EP  - 464
VL  - 49
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a5/
LA  - en
ID  - KYB_2013_49_3_a5
ER  - 
%0 Journal Article
%A Mordukhovich, Boris S.
%A Outrata, Jiří V.
%T Tilt stability in nonlinear programming under Mangasarian-Fromovitz constraint qualification
%J Kybernetika
%D 2013
%P 446-464
%V 49
%N 3
%U http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a5/
%G en
%F KYB_2013_49_3_a5
Mordukhovich, Boris S.; Outrata, Jiří V. Tilt stability in nonlinear programming under Mangasarian-Fromovitz constraint qualification. Kybernetika, Tome 49 (2013) no. 3, pp. 446-464. http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a5/

[1] Artacho, F. J. A. Aragón, Goeffroy, M. H.: Characterization of metric regularity of subdifferentials. J. Convex Anal. 15 (2008), 365-380. | MR

[2] Bonnans, F. J., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York 2000. | MR | Zbl

[3] Dontchev, A. L., Rockafellar, R. T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6 (1996), 1087-1105. | DOI | MR | Zbl

[4] Dontchev, A. L., Rockafellar, R. T.: Characterizations of Lipschitzian stability in nonlinear programming. In: Mathematical Programming with Data Perturbations (A. V. Fiacco, ed.), Marcel Dekker, New York 1997, pp. 65-82. | MR | Zbl

[5] Dontchev, A. L., Rockafellar, R. T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer, Dordrecht 2009. | MR | Zbl

[6] Drusvyatskiy, D., Lewis, A. S.: Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential. SIAM J. Optim. 23 (2013), 256-267. | DOI | MR

[7] Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York 2003. | Zbl

[8] Henrion, R., Mordukhovich, B. S., Nam, N. M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20 (2010), 2199-2227. | DOI | MR | Zbl

[9] Henrion, R., Outrata, J. V., Surowiec, T.: On the coderivative of normal cone mappings to inequality systems. Nonlinear Anal. 71 (2009), 1213-1226. | DOI | MR

[10] Henrion, R., Outrata, J. V., Surowiec, T.: On regular coderivatives in parametric equalibria with non-unique multipliers. Math. Programming Ser. B 136 (2012), 111-131. | DOI | MR

[11] Henrion, R., Kruger, A. Y., Outrata, J. V.: Some remarks on stability of generalized equations. J. Optim. Theory Appl., DOI 10.1007 s 10957-012-0147-x.

[12] Izmailov, A. F., Kurennoy, A. S., Solodov, M. V.: A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz continuous KKT systems. Math. Programming, DOI 10.1007/s 10107-012-0586-z.

[13] Janin, R.: Directional derivative of marginal function in nonlinear programming. Math. Programming Stud. 21 (1984), 110-126. | DOI | MR

[14] Klatte, D.: On the stability of local and global solutions in parametric problems of nonlinear programming. Part I: Basic results. Seminarbericht 75 der Sektion Mathematik der Humboldt-Universitat zu Berlin 1985, pp. 1-21, | MR

[15] Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer, Boston 2002. | MR | Zbl

[16] Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Analysis and Computation of Fixed Points (S. M. Robinson, ed.), Academic Press, New York 1980, pp. 93-138. | MR | Zbl

[17] Levy, A. B., Poliquin, R. A., Rockafellar, R. T.: Stability of local optimal solutions. SIAM J. Optim. 10 (2000), 580-604. | DOI | MR

[18] Lewis, A. S., Zhang, S.: Partial smoothness, tilt stability, and generalized Hessians. SIAM J. Optim. 23 (2013), 74-94. | DOI | MR

[19] Lu, S.: Implications of the constant rank constraint qualification. Math. Programming 126 (2011), 365-392. | DOI | MR | Zbl

[20] Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21 (2011), 314-332. | DOI | MR | Zbl

[21] Mordukhovich, B. S.: Sensitivity analysis in nonsmooth optimization. In: Theoretical Aspects of Industrial Design (D. A. Field and V. Komkov, eds.), SIAM Proc. Appl. Math. 58 (1992), pp. 32-46. Philadelphia. | MR | Zbl

[22] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, II: Applications. Springer, Berlin 2006. | MR | Zbl

[23] Mordukhovich, B. S., Outrata, J. V.: Second-order subdifferentials and their applications. SIAM J. Optim. 12 (2001), 139-169. | DOI | MR | Zbl

[24] Mordukhovich, B. S., Outrata, J. V.: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18 (2007), 389-412. | DOI | MR | Zbl

[25] Mordukhovich, B. S., Rockafellar, R. T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22 (2012), 953-986. | DOI | MR | Zbl

[26] Outrata, J. V.: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24 (1999), 627-644. | DOI | MR | Zbl

[27] Outrata, J. M., C., H. Ramírez: On the Aubin property of critical points to perturbed second-order cone programs. SIAM J. Optim. 21 (2011), 798-823. | DOI | MR | Zbl

[28] Poliquin, R. A., Rockafellar, R. T.: Tilt stability of a local minimum. SIAM J. Optim. 8 (1998), 287-299. | DOI | MR | Zbl

[29] Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Programming 70 (1995), 159-172. | DOI | MR | Zbl

[30] Robinson, S. M.: Generalized equations and their solutions, I: Basic theory. Math. Programming Stud. 10 (1979), 128-141. | DOI | MR | Zbl

[31] Robinson, S. M.: Strongly regular generalized equations. Math. Oper. Res. 5 (1980), 43-62. | DOI | MR | Zbl

[32] Robinson, S. M.: Local epi-continuity and local optimization. Math. Programming 37 (1987), 208-223. | DOI | MR | Zbl

[33] Rockafellar, R. T., Wets, R. J.-B.: Variational Analysis. Springer, Berlin 1998. | MR | Zbl