Startseite Mathematik Derivatives of Hadamard type in vector optimization
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Derivatives of Hadamard type in vector optimization

  • Karel Pastor EMAIL logo
Veröffentlicht/Copyright: 31. März 2018
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 68 Heft 2

Abstract

In our paper we will continue the comparison which was started by Vsevolod I. Ivanov [Nonlinear Analysis 125 (2015), 270–289], where he compared scalar optimality conditions stated in terms of Hadamard derivatives for arbitrary functions and those which was stated for ℓ-stable functions in terms of Dini derivatives. We will study the vector optimization problem and we show that also in this case the optimality condition stated in terms of Hadamard derivatives is more advantageous.

Keywords: C1,1-function; ℓ-stable function; generalized second-order derivative; optimality conditions
MSC 2010: 49K10; 49J52; 49J50; 90C29; 90C30

Communicated by Anatolij Dvurečenskij

References

[BP1] Bednařík, D.—Pastor, K.: On second-order conditions in unconstrained optimization, Math. Program. (Ser. A) 113 (2008), 283–298.10.1007/s10107-007-0094-8Suche in Google Scholar

[BP2] Bednařík, D.—Pastor, K.: Differentiability properties of functions that are ℓ-stable at a point, Nonlinear Anal. 69 (2008), 3128–3135.10.1016/j.na.2007.09.006Suche in Google Scholar

[BP3] Bednařík, D.—Pastor, K.: ℓ-stable functions are continuous, Nonlinear Anal. 70 (2009), 2317–2324.10.1016/j.na.2008.03.011Suche in Google Scholar

[BP4] Bednařík, D.—Pastor, K.: Fréchet approach in second-order optimization, Appl. Math. Lett. 22 (2009), 960–967.10.1016/j.aml.2009.01.019Suche in Google Scholar

[BP5] Bednařík, D.—Pastor, K.: ℓ-stable functions are continuous, Nonlinear Anal. 70 (2009), 2317–2324.10.1016/j.na.2008.03.011Suche in Google Scholar

[BP6] Bednařík, D.—Pastor, K.: Decrease of C1,1 property in vector optimization, RAIRO – Operations Research 43 (2009), 359–372.10.1051/ro/2009023Suche in Google Scholar

[BP7] Bednařík, D.—Pastor, K.: On l-stable mappings with values in infinite dimensional Banach spaces, Nonlinear Anal. 72 (2010), 1198–1209.10.1016/j.na.2009.08.004Suche in Google Scholar

[BP8] Bednařík, D.—Pastor, K.: On second-order condition in constrained vector optimization, Nonlinear Anal. 74 (2011), 1372–1382.10.1016/j.na.2010.10.009Suche in Google Scholar

[CC] Cominetti, R.—Correa, R.: A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim. 28 (1990), 789–809.10.1137/0328045Suche in Google Scholar

[CHN] Chan, W. L.—Huang, L. R.—Ng, K. F.: On generalized second-order derivatives and Taylor expansions in nonsmooth optimization, SIAM J. Control Optim. 32 (1994), 591–611.10.1137/S0363012992227423Suche in Google Scholar

[GGR] Ginchev, I.—Guerraggio, A.—Rocca, M.: From scalar to vector optimization, Appl. Math. 51 (2006), 5–36.10.1007/s10492-006-0002-1Suche in Google Scholar

[GL] Guerragio, A.—Luc, D. T.: Optimality conditions for C1,1 vector optimization problems, J. Optim. Theory Appl. 109 (2001), 615–629.10.1023/A:1017519922669Suche in Google Scholar

[GZ] Georgiev, P. G.—Zlateva, N. P.: Second-order Subdifferentials of C1,1 Functions and Optimality Conditions, Set-Valued Anal. 4 (1996), 101–117.10.1007/BF00425960Suche in Google Scholar

[DP1] Dvorská, M.—Pastor, K.: On comparison of ℓ-stable vector optimization results, Math. Slovaca 64 (2014), 1–18.10.2478/s12175-014-0252-4Suche in Google Scholar

[DP2] Dvorská, M.—Pastor, K.: Generalization of C1,1 property in infinite dimension, Acta Math. Vietnam. 41 (2016), 265–275.10.1007/s40306-015-0129-9Suche in Google Scholar

[DR] Demyanov, V. F.—Rubinov, M.: Constructive Nonsmooth Analysis. In: Approximation & Optimization, vol. 7, Peter Lang, Frankfurt am Main, 1995.Suche in Google Scholar

[G] Ginchev, I.: On scalar and vector ℓ-stable functions, Nonlinear Anal. 74 (2011), 182–194.10.1016/j.na.2010.08.032Suche in Google Scholar

[GG] Ginchev, I.—Guerraggio, A.: Second-order conditions for constrained vector optimization problems with ℓ-stable data, Optimization 60 (2011), 179–199.10.1080/02331930903578718Suche in Google Scholar

[HSN] Hiriart-Urruty, J. B.—Strodiot, J. J.—Nguyen, V. H.: Generalized Hessian matrix and second-order optimality conditions for problems with C1,1 data, Appl. Math. Optim. 11 (1984), 43–56.10.1007/BF01442169Suche in Google Scholar

[I] Ivanov, V. I.: Second-optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization, Nonlinear Anal. 125 (2015), 270–289.10.1016/j.na.2015.05.030Suche in Google Scholar

[J] Jahn, J.: Vector Optimization, Springer-Verlag, New York, 2004.10.1007/978-3-540-24828-6Suche in Google Scholar

[JL] Jeyakumar, V.—Luc, D. T.: Nonsmooth Vector Functions and Continuous Optimization, Springer, Berlin, 2008.Suche in Google Scholar

[KT1] Khanh, P. Q.—Tuan, N. D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives, J. Optim. Theory Appl. 133 (2007), 341–357.10.1007/s10957-007-9169-1Suche in Google Scholar

[KT2] Khanh, P. Q.—Tuan, N. D.: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming I: ℓ-stability and set-valued directional derivatives, J. Math. Anal. Appl. 403 (2013), 695–702.10.1016/j.jmaa.2012.12.076Suche in Google Scholar

[KT3] Khanh, P. Q.—Tuan, N. D.: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming II: Optimality conditions, J. Math. Anal. Appl. 403 (2013), 703–714.10.1016/j.jmaa.2012.12.075Suche in Google Scholar

[LK] Liu, L.—Křížek, M.: The second-order optimality conditions for nonlinear mathematical programming with C1,1 data, Appl. Math. 42 (1997), 311–320.10.1023/A:1023068513188Suche in Google Scholar

[LNK] Liu, L.—Neittaanmäki, P.—Křížek, M.: Second-order optimality conditions for nondominated solutions of multiobjective programming with C1,1 data, Appl. Math. 45 (2000), 381–397.10.1023/A:1022272728208Suche in Google Scholar

[LX] Li, S. J.—Xu, S.: Sufficient conditions of isolated minimizers for constrained programming problems, Numer. Funct. Anal. Optim. 31 (2010), 715–727.10.1080/01630563.2010.490970Suche in Google Scholar

[M] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Springer, Berlin, 2006.10.1007/3-540-31246-3Suche in Google Scholar

[P1] Pastor, K.: Differentiability properties of ℓ-stable vector functions in infinite-dimensional normed spaces, Taiwanese J. Math. 18 (2014), 187–197.10.11650/tjm.18.2014.2605Suche in Google Scholar

[P2] Pastor, K.: A note on second-order optimality conditions, Nonlinear Anal. 71 (2009), 1964–1969.10.1016/j.na.2009.01.035Suche in Google Scholar

[R] Rockafellar, R. T.: Convex Analysis, Princeton University Press, Princeton, 1970.10.1515/9781400873173Suche in Google Scholar

[RW] Rockafellar, R. T.—Wets, R. J.-B.: Variational Analysis, Springer-Verlag, New York, 1998.10.1007/978-3-642-02431-3Suche in Google Scholar

[TR] Torre, D. L.—Rocca, M.: Remarks on second order generalized derivatives for differentiable functions with Lipschitzian jacobian, Appl. Math. E-Notes 3 (2003), 130–137.Suche in Google Scholar

[Y1] Yang, X. Q.: On second-order directional derivatives, Nonlinear Anal. 26 (1996), 55–66.10.1016/0362-546X(94)00209-ZSuche in Google Scholar

[Y2] Yang, X. Q.: On relations and applications of generalized second-order directional derivatives, Nonlinear Anal. 36 (1999), 595–614.10.1016/S0362-546X(98)00174-6Suche in Google Scholar

Received: 2016-1-12
Accepted: 2016-9-26
Published Online: 2018-3-31
Published in Print: 2018-4-25

© 2018 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  2. On the family of functions with closure of graphs in the Mendez ideals
  4. The predicate completion of a partial information system
  6. On lattices of z-ideals of function rings
  8. Compactifications of partial frames via strongly regular ideals
  10. Lifting components in clean abelian ℓ-groups
  12. Invariance of nonatomic measures on effect algebras
  14. On the alexander dual of path ideals of cycle posets
  16. Generalized multiplicative derivations in 3-prime near rings
  18. Cleft extensions for quasi-entwining structures
  20. Groups with positive rank gradient and their actions
  22. Certain results on q-starlike and q-convex error functions
  24. Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate
  26. Positive periodic solutions for singular high-order neutral functional differential equations
  28. A special class of functional equations
  30. Matrix mappings and general bounded linear operators on the space bv
  32. Skew-symmetric operators and reflexivity
  34. Derivatives of Hadamard type in vector optimization
  36. Cardinal functions of the hyperspace of convergent sequences
  38. Suzuki-type of common fixed point theorems in fuzzy metric spaces
  40. Second hankel determinat for certain analytic functions satisfying subordinate condition
https://doi.org/10.1515/ms-2017-0113
Schlagwörter für diesen Artikel
C1,1-function; ℓ-stable function; generalized second-order derivative; optimality conditions

Artikel in diesem Heft

  2. On the family of functions with closure of graphs in the Mendez ideals
  4. The predicate completion of a partial information system
  6. On lattices of z-ideals of function rings
  8. Compactifications of partial frames via strongly regular ideals
  10. Lifting components in clean abelian ℓ-groups
  12. Invariance of nonatomic measures on effect algebras
  14. On the alexander dual of path ideals of cycle posets
  16. Generalized multiplicative derivations in 3-prime near rings
  18. Cleft extensions for quasi-entwining structures
  20. Groups with positive rank gradient and their actions
  22. Certain results on q-starlike and q-convex error functions
  24. Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate
  26. Positive periodic solutions for singular high-order neutral functional differential equations
  28. A special class of functional equations
  30. Matrix mappings and general bounded linear operators on the space bv
  32. Skew-symmetric operators and reflexivity
  34. Derivatives of Hadamard type in vector optimization
  36. Cardinal functions of the hyperspace of convergent sequences
  38. Suzuki-type of common fixed point theorems in fuzzy metric spaces
  40. Second hankel determinat for certain analytic functions satisfying subordinate condition
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0113/pdf?lang=de
Button zum nach oben scrollen