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Keywords:
variational analysis; second-order theory; generalized differentiation; tilt stability
variational analysis; second-order theory; generalized differentiation; tilt stability
Summary:
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.
References:
[1] Artacho, F. J. A. Aragón, Goeffroy, M. H.: Characterization of metric regularity of subdifferentials. J. Convex Anal. 15 (2008), 365-380. MR 2422996
[2] Bonnans, F. J., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York 2000. MR 1756264 | Zbl 0966.49001
[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 10.1137/S1052623495284029 | MR 1416530 | Zbl 0899.49004
[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 1472266 | Zbl 0891.90146
[5] Dontchev, A. L., Rockafellar, R. T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer, Dordrecht 2009. MR 2515104 | Zbl 1178.26001
[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 10.1137/120876551 | MR 3033107
[7] Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York 2003. Zbl 1062.90002
[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 10.1137/090766413 | MR 2650845 | Zbl 1208.49010
[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 10.1016/j.na.2008.11.089 | MR 2527541
[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 10.1007/s10107-012-0553-8 | MR 3000584
[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 10.1007/BFb0121214 | MR 0751246
[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 0861527
[15] Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer, Boston 2002. MR 1909427 | Zbl 1173.49300
[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 0592631 | Zbl 0478.90062
[17] Levy, A. B., Poliquin, R. A., Rockafellar, R. T.: Stability of local optimal solutions. SIAM J. Optim. 10 (2000), 580-604. DOI 10.1137/S1052623498348274 | MR 1740960
[18] Lewis, A. S., Zhang, S.: Partial smoothness, tilt stability, and generalized Hessians. SIAM J. Optim. 23 (2013), 74-94. DOI 10.1137/110852103 | MR 3033099
[19] Lu, S.: Implications of the constant rank constraint qualification. Math. Programming 126 (2011), 365-392. DOI 10.1007/s10107-009-0288-3 | MR 2764353 | Zbl 1214.90113
[20] Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21 (2011), 314-332. DOI 10.1137/090761318 | MR 2783218 | Zbl 1229.90216
[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 1157413 | Zbl 0769.90075
[22] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, II: Applications. Springer, Berlin 2006. MR 2191744 | Zbl 1100.49002
[23] Mordukhovich, B. S., Outrata, J. V.: Second-order subdifferentials and their applications. SIAM J. Optim. 12 (2001), 139-169. DOI 10.1137/S1052623400377153 | MR 1870589 | Zbl 1011.49016
[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 10.1137/060665609 | MR 2338444 | Zbl 1145.49012
[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 10.1137/110852528 | MR 3023759 | Zbl 1260.49022
[26] Outrata, J. V.: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24 (1999), 627-644. DOI 10.1287/moor.24.3.627 | MR 1854246 | Zbl 1039.90088
[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 10.1137/100807168 | MR 2837552 | Zbl 1247.90256
[28] Poliquin, R. A., Rockafellar, R. T.: Tilt stability of a local minimum. SIAM J. Optim. 8 (1998), 287-299. DOI 10.1137/S1052623496309296 | MR 1618790 | Zbl 0918.49016
[29] Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Programming 70 (1995), 159-172. DOI 10.1007/BF01585934 | MR 1361325 | Zbl 0844.90089
[30] Robinson, S. M.: Generalized equations and their solutions, I: Basic theory. Math. Programming Stud. 10 (1979), 128-141. DOI 10.1007/BFb0120850 | MR 0527064 | Zbl 0404.90093
[31] Robinson, S. M.: Strongly regular generalized equations. Math. Oper. Res. 5 (1980), 43-62. DOI 10.1287/moor.5.1.43 | MR 0561153 | Zbl 0437.90094
[32] Robinson, S. M.: Local epi-continuity and local optimization. Math. Programming 37 (1987), 208-223. DOI 10.1007/BF02591695 | MR 0883021 | Zbl 0623.90078
[33] Rockafellar, R. T., Wets, R. J.-B.: Variational Analysis. Springer, Berlin 1998. MR 1491362 | Zbl 0888.49001
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