Geodesic
On second–order Taylor expansion of critical values
Kybernetika, Tome 46 (2010) no. 3, pp. 472-487 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Studying a critical value function $\varphi$ in parametric nonlinear programming, we recall conditions guaranteeing that $\varphi$ is a $C^{1,1}$ function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of $D \varphi$. Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization.
Studying a critical value function $\varphi$ in parametric nonlinear programming, we recall conditions guaranteeing that $\varphi$ is a $C^{1,1}$ function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of $D \varphi$. Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization.
Classification : 49J52, 49K40, 65K05, 65K10, 90C30, 90C31
Keywords: Taylor expansion; parametric programs; critical value function; generalized derivatives; envelope theorems; Lipschitz stability; $C^{1, 1}$ optimization 
@article{KYB_2010_46_3_a11,
     author = {B\"utikofer, Stephan and Klatte, Diethard and Kummer, Bernd},
     title = {On second{\textendash}order {Taylor} expansion of critical values},
     journal = {Kybernetika},
     pages = {472--487},
     year = {2010},
     volume = {46},
     number = {3},
     mrnumber = {2676084},
     zbl = {1197.65062},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a11/}
}
TY  - JOUR
AU  - Bütikofer, Stephan
AU  - Klatte, Diethard
AU  - Kummer, Bernd
TI  - On second–order Taylor expansion of critical values
JO  - Kybernetika
PY  - 2010
SP  - 472
EP  - 487
VL  - 46
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a11/
LA  - en
ID  - KYB_2010_46_3_a11
ER  - 
%0 Journal Article
%A Bütikofer, Stephan
%A Klatte, Diethard
%A Kummer, Bernd
%T On second–order Taylor expansion of critical values
%J Kybernetika
%D 2010
%P 472-487
%V 46
%N 3
%U http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a11/
%G en
%F KYB_2010_46_3_a11
Bütikofer, Stephan; Klatte, Diethard; Kummer, Bernd. On second–order Taylor expansion of critical values. Kybernetika, Tome 46 (2010) no. 3, pp. 472-487. http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a11/

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

[2] Bütikofer, St.: Globalizing a nonsmooth Newton method via nonmonotone path search. Math. Meth. Oper. Res. 68 (2008), 235–256. | DOI | MR

[3] Bütikofer, St., Klatte, D.: A nonsmooth Newton method with path search and its use in solving $C^{1,1}$ programs and semi-infinite problems. Manuscript, February 2009.

[4] Clarke, F. H.: Optimization and Nonsmooth Analysis. Wiley, New York 1983. | MR | Zbl

[5] Dempe, S.: Foundations of Bilevel Programming. Kluwer, Dordrecht – Boston – London 2002. | MR | Zbl

[6] Demyanov, V. F., Malozemov, V. N.: Introduction to Minimax. Wiley, New York 1974. | MR

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

[8] Fiacco, A. V.: Introduction to Sensitivity and Stability Analysis. Academic Press, New York 1983. | MR | Zbl

[9] Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Study 19 (1982), 101–119. | DOI | MR | Zbl

[10] Gauvin, J.: Theory of Nonconvex Programming. Les Publications CRM, Montreal 1994.

[11] Golstein, E. G.: Theory of Convex Programming. (Trans. Math. Monographs 36.) American Mathematical Society, Providence 1972. | MR

[12] Hiriart-Urruty, J.-B., Strodiot, J. J., Nguyen, V. Hien: Generalized Hessian matrix and second order optimality conditions for problems with $ {C}^{1,1} $-data. Appl. Math. Optim. 11 (1984), 43–56. | DOI | MR

[13] Jittorntrum, K.: Solution point differentiability without strict complementarity in nonlinear programming. Math. Program. Study 21 (1984), 127–138. | DOI | MR | Zbl

[14] Jongen, H. Th., Möbert, T., Tammer, K.: On iterated minimization in nonconvex optimization. Math. Oper. Res. 11 (1986), 679–691. | DOI | MR

[15] Klatte, D.: On quantitative stability for non-isolated minima. Control and Cybernetics 23 (1994), 183–200. | MR | Zbl

[16] Klatte, D., Kummer, B.: Generalized Kojima functions and Lipschitz stability of critical points. Comput. Optim. Appl. 13 (1999), 61–85. | DOI | MR | Zbl

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

[18] Klatte, D., Kummer, B.: Optimization methods and stability of inclusions in Banach spaces. Math. Program. Ser. B 117 (2009), 305–330. | DOI | MR | Zbl

[19] Klatte, D., Tammer, K.: Strong stability of stationary solutions and Karush–Kuhn–Tucker points in nonlinear optimization. Ann. Oper. Res. 27 (1990), 285–307. | DOI | MR | Zbl

[20] 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

[21] Kummer, B.: Newton’s method for non-differentiable functions. In: Advances in Math. Optimization (J. Guddat et al., eds.), Akademie Verlag, Berlin 1988, pp. 114–125. | Zbl

[22] Kummer, B.: Lipschitzian inverse functions, directional derivatives and application in $ {C}^{1,1} $ optimization. J. Optim. Theory Appl. 70 (1991), 559–580. | DOI | MR

[23] Kummer, B.: An implicit function theorem for $ {C}^{0,1} $-equations and parametric $ {C}^{1,1} $-optimization. J. Math. Anal. Appl. 158 (1991), 35–46. | DOI | MR

[24] Kummer, B.: Newton’s method based on generalized derivatives for nonsmooth functions: convergence analysis. In: Advances in Optimization (W. Oettli and D. Pallaschke, eds.), Springer, Berlin 1992, pp. 171–194. | MR | Zbl

[25] Kummer, B.: Generalized Newton and NCP-methods: Convergence, regularity, actions. Discuss. Math. – Differential Inclusions 20 (2000), 209–244. | DOI | MR | Zbl

[26] Kummer, B.: Inclusions in Banach spaces: Hoelder stability, solution schemes and Ekeland’s principle. J. Math. Anal. Appl. 358 (2009), 327–344. | DOI | MR

[27] Minchenko, L. I.: Multivalued analysis and differential properties of multivalued mappings and marginal functions. J. Math. Sci. 116 (2003), 93–138. | DOI | MR

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

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

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

[31] Scholtes, S.: Introduction to Piecewise Differentiable Equations. Preprint No. 53/1994. Institut für Statistik und Math. Wirtschaftstheorie, Universität Karlsruhe, 1994.

[32] Stein, O.: Bi-level Strategies in Semi-infinite Programming. Kluwer, Dordrecht – Boston – London 2003. | MR | Zbl

[33] Sydsaeter, K., Hammond, P., Seierstad, A., Strom, A.: Further Mathematics for Economic Analysis. Prentice Hall, 2005.

[34] Thibault, L.: Subdifferentials of compactly Lipschitz vector-valued functions. Ann. Mat. Pura Appl. 4 (1980), 157–192. | DOI | MR

[35] Varian, H.: Microeconomic Analysis. Third edition. W. W. Norton, New York 1992.