Set-Valued and Variational Analysis

, Volume 26, Issue 4, pp 911–946 | Cite as

Full Stability of General Parametric Variational Systems

  • B. S. MordukhovichEmail author
  • T. T. A. Nghia
  • D. T. Pham


The paper introduces and studies the notions of Lipschitzian and Hölderian full stability of solutions to three-parametric variational systems described in the generalized equation formalism involving nonsmooth base mappings and partial subgradients of prox-regular functions acting in Hilbert spaces. Employing advanced tools and techniques of second-order variational analysis allows us to establish complete characterizations of, as well as directly verifiable sufficient conditions for, such full stability properties under mild assumptions. Furthermore, we derive exact formulas and effective quantitative estimates for the corresponding moduli. The obtained results are specified for important classes of variational inequalities and variational conditions in both finite and infinite dimensions.


Variational analysis Parametric variational systems Variational inequalities and variational conditions Lipschitzian and Hölderian full stability Prox-regularity Legendre forms Polyhedricity Generalized differentiation Subgradients Coderivatives 

Mathematics Subject Classification (2010)

Primary 49J53 Secondary 49J52, 90C31 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bayen, T., Bonnans, J.F., Silva, F.J.: Characterization of local quadratic growth for strong minima in the optimal control of semilinear elliptic equations. Trans. Amer. Math. Soc. 366, 2063–2087 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303, 1–14 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bonnans, J.F.: Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38, 303–325 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefGoogle Scholar
  5. 5.
    Chieu, N.H., Trang, N.T.Q.: Coderivative and monotonicity of continuous mappings. Taiwan. J. Math. 16, 353–365 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dafermos, S.: Sensitivity analysis in variational inequalities. Math. Oper. Res. 13, 421–434 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2nd edn. Springer, New York (2014)zbMATHGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J.-S.: Finite-Dimesional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
  10. 10.
    Haraux, A.: How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29, 615–707 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Henrion, R., Kruger, A.Y., Outrata, J.V.: Some remarks on stability of generalized equations. J. Optim. Theory Appl. 159, 681–697 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hintermüller, M., Surowiec, T.: First-order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysis. SIAM J. Optim. 21, 1561–1593 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hintermüller, M., Mordukhovich, B.S., Surowiec, T.: Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. Math. Program. 146, 555–582 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979)Google Scholar
  15. 15.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  16. 16.
    Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)CrossRefGoogle Scholar
  17. 17.
    Kyparisis, J.: Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers. Math. Oper. Res. 15, 286–298 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities. A Qualitative Study. Springer, New York (2006)zbMATHGoogle Scholar
  19. 19.
    Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lu, S.: Variational conditions under the constant rank constraint qualification. Math. Oper. Res. 35, 120–139 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lu, S., Robinson, S.M.: Variational inequalities over perturbed polyhedral convex sets. Math. Oper. Res. 33, 689–711 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mignot, F.: Contrôle dans les inéquations variationnelles elliptiques. J. Func. Anal. 22, 25–39 (1976)CrossRefGoogle Scholar
  23. 23.
    Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, SIAM Proc. Applied Math., vol. 58, pp. 32–46 (1992)Google Scholar
  24. 24.
    Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340, 1–36 (1993)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory, II: Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  26. 26.
    Mordukhovich, B.S., Nghia, T.T.A.: Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces. Nonlinear Anal. 86, 159–180 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mordukhovich, B.S., Nghia, T.T.A.: Full Lipschitzian and Hölderian stability in optimization with applications to mathematical programming and optimal control. SIAM J. Optim. 24, 1344–1381 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mordukhovich, B.S., Nghia, T.T.A.: Second-order characterizations of tilt stability with applications to nonlinear programming. Math. Program. 149, 83–104 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mordukhovich, B.S., Nghia, T.T.A.: Local monotonicity and full stability for parametric variational systems. SIAM J. Optim. 26, 1032–1059 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasi-variational inequalities with applications to stability in optimization. SIAM J. Optim. 18, 389–412 (2007)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Mordukhovich, B.S., Nam, N.M., Nhi, N.T.Y.: Partial second-order subdifferentials in variational analysis and optimization. Numer. Func. Anal. Optim. 35, 1113–1151 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mordukhovich, B.S., Nghia, T.T.A., Rockafellar, R.T.: Full stability in finite-dimensional optimization. Math. Oper. Res. 40, 226–252 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 170–191 (2002)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Poliquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348, 1805–1838 (1996)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Robinson, S.M.: Generalized equations and their solutions, I: basic theory. Math. Program. Stud. 10, 128–141 (1979)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Robinson, S.M.: Generalized equations and their solutions, II: applications to nonlinear programming. Math. Program. Stud. 19, 200–221 (1982)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Robinson, S.M.: Variational conditions with smooth constraints. Structure and analysis. Math. Program. 97, 245–265 (2003)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Robinson, S.M.: Aspects of the projector on prox-regular sets. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications, pp. 963–973. Springer, New York (2005)Google Scholar
  43. 43.
    Robinson, S.M.: Equations on monotone graphs. Math. Program. 141, 49–101 (2013)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Sebbah, M., Thibault, L.: Metric projection and compatibly parameterized families of prox-regular sets in Hilbert space. Nonlinear Anal. 75, 1547–1562 (2012)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar
  46. 46.
    Stampacchia, G.: Formes bilinéaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Walkup, D.W., Wets, R.J.-B.: A Lipschitzian characterization of convex polyhedra. Proc. Amer. Math. Soc. 20, 167–173 (1969)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Yen, N.D.: Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint. Math. Oper. Res. 20, 695–708 (1995)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Yen, N.D.: Hölder continuity of solutions to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • B. S. Mordukhovich
    • 1
    • 2
    Email author
  • T. T. A. Nghia
    • 3
  • D. T. Pham
    • 1
  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.RUDN UniversityMoscowRussia
  3. 3.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

Personalised recommendations