Differential Stability of Convex Optimization Problems with Possibly Empty Solution Sets

  • Duong Thi Viet An
  • Jen-Chih YaoEmail author


This paper studies differential stability of infinite-dimensional convex optimization problems, whose solution sets may be empty. By using suitable sum rules for \(\varepsilon \)-subdifferentials, we obtain exact formulas for computing the \(\varepsilon \)-subdifferential of the optimal value function. Several illustrative examples are also given.


Parametric convex programming Optimal value function Conjugate function \(\varepsilon \)-Subdifferentials \(\varepsilon \)-Normal directions 

Mathematics Subject Classification

49J53 49Q12 90C25 90C31 



The research of Duong Thi Viet An was supported by Thai Nguyen University of Sciences and the Vietnam Institute for Advanced Study in Mathematics (VIASM). The research of Jen-Chih Yao was supported by the Grant MOST 105-2221-E-039-009-MY3. The authors would like to thank Prof. Nguyen Dong Yen for useful comments and suggestions.


  1. 1.
    An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal. 94, 108–128 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    An, D.T.V., Yao, J.-C.: Further results on differential stability of convex optimization problems. J. Optim. Theory Appl. 170, 28–42 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Volume I: Basic Theory, Volume II: Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  4. 4.
    Mordukhovich, B.S., Nam, N.M., Rector, B., Tran, T.: Variational geometric approach to generalized differential and conjugate calculi in convex analysis. Set Valued Var. Anal. 25, 731–755 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B 116, 369–396 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Penot, J.-P.: Calculus Without Derivatives. Graduate Texts in Mathematics. Springer, New York (2013)CrossRefGoogle Scholar
  7. 7.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  8. 8.
    Zănlinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, London (2002)CrossRefGoogle Scholar
  9. 9.
    Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Volle, M.: Calculus rules for global approximate minima and applications to approximate subdifferential calculus. J. Glob. Optim. 5, 131–157 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hiriart-Urruty, J.-B.: \(\varepsilon \)-subdifferential calculus. Convex analysis and optimization. Res. Notes in Math., Vol. 57, pp. 43–92. Pitman, Boston (1982)Google Scholar
  12. 12.
    Hiriart-Urruty, J.-B.: From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality. In: Clarke, F.H., Demyanov, V.F., Giannessi, F. (eds.) Nonsmooth Optimization and Related Topics. Ettore Majorana Internat. Sci. Ser. Phys. Sci., Vol. 43, pp. 219–239. Plenum, New York (1989)Google Scholar
  13. 13.
    Hiriart-Urruty, J.-B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 24, 1727–1754 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Aalgorithms. II. Advanced Theory and Bundle Methods. Grundlehren Math. Wiss. Springer, Berlin (1993)zbMATHGoogle Scholar
  15. 15.
    Hiriart-Urruty, J.-B., Phelps, R.R.: Subdifferential calculus using \(\varepsilon \)-subdifferentials. J. Funct. Anal. 118, 154–166 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moussaoui, M., Seeger, A.: Sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets. SIAM J. Optim. 4, 659–675 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Seeger, A.: Approximate Euler-Lagrange inclusion, approximate transversality condition, and sensitivity analysis of convex parametric problems of calculus of variations. Set Valued Anal. 2, 307–325 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Seeger, A.: Subgradients of optimal-value functions in dynamic programming: the case of convex systems without optimal paths. Math. Oper. Res. 21, 555–575 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)zbMATHGoogle Scholar
  20. 20.
    Attouch, H., Brezis, H.: Duality for the sum of convex functions in general Banach spaces. In: Barraso, J.A. (ed.) Aspects of Mathematics and Its Applications, vol. 34, pp. 125–133. North-Holland Mathematical Library, Amsterdam (1986)CrossRefGoogle Scholar
  21. 21.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsThai Nguyen University of SciencesThai Nguyen CityVietnam
  2. 2.Center for General EducationChina Medical UniversityTaichungTaiwan

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