[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We consider higher-order conditions and sensitivity analysis for solutions to equilibrium problems. The conditions for solutions are in terms of quasi-contingent derivatives and involve higher-order complementarity slackness for both the objective and the constraints and under Hölder metric subregularity assumptions. For sensitivity analysis, a formula of this type of derivative of the solution map to a parametric equilibrium problem is established in terms of the same types of derivatives of the data of the problem. Here, the concepts of a quasi-contingent derivative and critical directions are new. We consider open-cone solutions and proper solutions. We also study an important and typical special case: weak solutions of a vector minimization problem with mixed constraints. The results are significantly new and improve recent corresponding results in many aspects.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anh, N.L.H., Khanh, P.Q.: Variational sets of perturbation maps and applications to sensitivity analysis for constrained vector optimization. J. Optim. Theory Appl. 158(2), 363–384 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh, N.L.H., Khanh, P.Q.: Higher-order conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives. J. Glob. Optim. 58(4), 693–709 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anh, N.L.H., Khanh, P.Q.: Calculus and applications of Studniarski’s derivative to sensitivity and implicit function theorems. Control Cybern. 43(1), 33–57 (2014)

  4. Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization. Nonlinear Anal. TMA 74(18), 7365–7379 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anitescu, M.: Degenerate nonlinear programming with a quadratic growth condition. SIAM J. Optim. 10(4), 1116–1135 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  7. Bednarczuk, E., Song, W.: Contingent epiderivative and its applications to set-valued optimization. Control Cybern. 24, 375–386 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  9. Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21(1), 265–287 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. de Araujo, A.P., Monteiro, P.K.: On programming when the positive cone has an empty interior. J. Optim. Theory Appl. 67(2), 395–410 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diem, H.T.H., Khanh, P.Q., Tung, L.T.: On higher-order sensitivity analysis in nonsmooth vector optimization. J. Optim. Theory Appl. 162(2), 463–488 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Durea, M., Dutta, J., Tammer, C.: Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension. J. Math. Anal. Appl. 348(2), 589–606 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gauvin, J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Program. 12(1), 136–138 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ginchev, I.: Higher order optimality conditions in nonsmooth optimization. Optimization 51(1), 47–72 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gollan, B.: Higher order necessary conditions for an abstract optimization problem. In: Mathematics Programming on Studying, pp. 69–76. Springer, Berlin (1981)

  16. Gong, X.H.: Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. 73(11), 3598–3612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Götz, A., Jahn, J.: The Lagrange multiplier rule in set-valued optimization. SIAM J. Optim. 10(2), 331–344 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. in: Pardalos, P., Rassis, Th.M., Khan, A.A. (Eds.) Nonlinear Analysis and Variational Problems, pp. 305–324 (Chapter 21). Springer (2009)

  19. Hoffmann, K.H., Kornstaedt, H.J.: Higher-order necessary conditions in abstract mathematical programming. J. Optim. Theory Appl. 26(4), 533–568 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ioffe, A.D.: Nonlinear regularity models. Math. Program. 139(1–2), 223–242 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ivanov, V.I.: Higher order optimality conditions for inequality-constrained problems. Appl. Anal. 92(12), 2600–2617 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems. Math. Program. 41(1–3), 73–96 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  24. Khanh, P.Q.: Proper solutions of vector optimization problems. J. Optim. Theory Appl. 74(1), 105–130 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Khanh, P.Q., Kruger, A.Y., Thao, N.H.: An induction theorem and nonlinear regularity models. SIAM J. Optim. 25(4), 2561–2588 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khanh, P.Q., Tuan, N.D.: Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization. J. Optim. Theory Appl. 139(2), 243–261 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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(2), 703–714 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Khanh, P.Q., Tung, L.T.: First and second-order optimality conditions using approximations for vector equilibrium problems with constraints. J. Glob. Optim. 55(4), 901–920 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khanh, P.Q., Tung, N.M.: Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints. Optimization 64(7), 1547–1575 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Khanh, P.Q., Tung, N.M.: Second-order optimality conditions with the envelope-like effect for set-valued optimization. J. Optim. Theory Appl. 167(1), 68–90 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Khanh, P.Q., Tung, N.M.: Existence and boundedness of second-order Karush–Kuhn–Tucker multipliers for set-valued optimization with variable ordering structures. Taiw. J. Math. 22(4), 1001–1029 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Khanh, P.Q., Tung, N.M.: Higher-order Karush–Kuhn–Tucker conditions in nonsmooth optimization. SIAM J. Optim. 28(1), 820–848 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ledzewicz, U., Schättler, H.: High-order approximations and generalized necessary conditions for optimality. SIAM J. Control Optim. 37(1), 33–53 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, G., Mordukhovich, B.S.: Hölder metric subregularity with applications to proximal point method. SIAM J. Optim. 22(4), 1655–1684 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, S.J., Li, M.H.: Sensitivity analysis of parametric weak vector equilibrium problems. J. Math. Anal. Appl. 380(1), 354–362 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, S.J., Teo, K.L., Yang, X.Q.: Higher-order optimality conditions for set-valued optimization. J. Optim. Theory Appl. 137(3), 533–553 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Luu, D.V.: Higher-order efficiency conditions via higher-order tangent cones. Numer. Funct. Anal. Appl. 35(1), 68–84 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Luu, D.V.: Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems. J. Glob. Optim. 70(2), 437–453 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ma, B.C., Gong, X.H.: Optimality conditions for vector equilibrium problems in normed spaces. Optimization 60(12), 1441–1455 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Makarov, E.K., Rachkovski, N.N.: Unified representation of proper efficiency by means of dilating cones. J. Optim. Theory Appl. 101(1), 141–165 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic theory, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  42. Mordukhovich, B.S., Ouyang, W.: Higher-order metric subregularity and its applications. J. Glob. Optim. 63(4), 777–795 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Páles, Z., Zeidan, V.M.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32(5), 1476–1502 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  44. Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37(1), 303–318 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. Penot, J.P.: Higher-order optimality conditions and higher-order tangent sets. SIAM J. Optim. 27(4), 2508–2527 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. Robinson, S.M.: Generalized equations and their solutions, I: Basic theory. In: Mathematics Programming Studies, pp. 128–141. Springer, Berlin (1979)

  47. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  48. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1985)

    MATH  Google Scholar 

  49. Studniarski, M.: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim. 24(5), 1044–1049 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  50. Studniarski, M.: Higher-order necessary optimality conditions in terms of Neustadt derivatives. Nonlinear Anal. 47(1), 363–373 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  51. Thibault, L.: Tangent cones and quasi-interiorly tangent cones to multifunctions. Trans. Am. Math. Soc. 277(2), 601–621 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zheng, X.Y., Ng, K.F.: The Fermat rule for multifunctions on Banach spaces. Math. Program. 104(1), 69–90 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work was supported by the National Foundation for Science and Technology Developments (NAFOSTED) of Vietnam under Grant 101.01-2021.13. A part of this work was completed during a scientific stay of the second and third authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). They would like to thank VIASM for its hospitality and support. The authors are warmly grateful to the handling editor and the anonymous referee for their helpful remarks and valuable suggestions.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

    Nguyen Xuan Duy Bao

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Nguyen Xuan Duy Bao

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Phan Quoc Khanh

  4. Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Minh Tung

Authors
  1. Nguyen Xuan Duy Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Phan Quoc Khanh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Minh Tung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Phan Quoc Khanh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bao, N.X.D., Khanh, P.Q. & Tung, N.M. Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization. J Glob Optim 84, 205–228 (2022). https://doi.org/10.1007/s10898-022-01129-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-022-01129-z

Keywords

Mathematics Subject Classification

Search

Navigation