Mathematical Programming

, Volume 169, Issue 2, pp 605–648 | Cite as

Critical multipliers in variational systems via second-order generalized differentiation

Full Length Paper Series A


In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush–Kuhn–Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal–dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness.


Variational systems Composite optimization Critical and noncritical multipliers Generalized differentiation Piecewise linear functions Robust isolated calmness Lipschitzian stability 

Mathematics Subject Classification

90C31 49J52 49J53 



The first author gratefully acknowledges numerous discussions with Alexey Izmailov and Mikhail Solodov on critical multipliers and related topics. We particularly appreciate sharing with us Izmailov’s instructive notes [17]. We are also indebted to two anonymous referees and the handling editor for their very careful reading of the paper and making helpful remarks that allowed us to improve the original presentation.


  1. 1.
    Bonnans, J.F.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefMATHGoogle Scholar
  3. 3.
    Chieu, N.H., Hien, L.V.: Computation of graphical derivative for a class of normal cone mappings under a very weak condition. SIAM J. Optim. 27, 190–204 (2017)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ding, C., Sun, D., Zhang, L.: Characterization of the robust isolated calmness for a class of conic programming problems. SIAM J. Optim. 27, 67–90 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of Lipschitzian stability in nonlinear programming. In: Fiacco, A.V. (ed.) Mathematical Programming with Data Perturbations, pp. 65–82. Marcel Dekker, New York (1997)Google Scholar
  6. 6.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd edn. Springer, New York (2014)MATHGoogle Scholar
  7. 7.
    Facchinei, F., Pang, J.-S.: Finite-Dimesional Variational Inequalities and Complementarity Problems. Springer, New York (2003)MATHGoogle Scholar
  8. 8.
    Fischer, A.: Local behavior of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94, 91–124 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fusek, P.: Isolated zeros of Lipschitzian metrically regular \({\mathbb{R}}^n\)-functions. Optimization 49, 425–446 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gfrerer, H.: First-order and second-order characterizations of metric subregularity and calmness of constraint mappings. SIAM J. Optim. 21, 1439–1474 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gfrerer, H., Mordukhovich, B.S.: Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions. SIAM J. Optim. 25, 2081–2119 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gfrerer, H., Mordukhovich, B.S.: Robinson stability of parametric constraint systems via variational analysis. SIAM J. Optim. 27, 438–465 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gfrerer, H., Outrata, J.V.: On computation of generalized derivatives of the normal cone mapping and their applications. Math. Oper. Res. 41, 1535–1556 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program. 104, 437–464 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16, 199–227 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Izmailov, A.F.: On the analytical and numerical stability of critical Lagrange multipliers. Comput. Math. Math. Phys. 45, 930–946 (2005)MathSciNetGoogle Scholar
  17. 17.
    Izmailov, A.F.: Tilt and full stability in constrained optimization and the existence of critical Lagrange multipliers, unpublished manuscript, (2015)Google Scholar
  18. 18.
    Izmailov, A.F., Solodov, M.V.: Stabilized SQP revisited. Math. Program. 133, 93–120 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, New York (2014)CrossRefMATHGoogle Scholar
  20. 20.
    Izmailov, A.F., Solodov, M.V.: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it. TOP 23, 1–26 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    King, A., Rockafellar, R.T.: Sensitivity analysis for nonsmooth generalized equations. Math. Oper. Res. 55, 341–364 (1992)MathSciNetMATHGoogle Scholar
  22. 22.
    Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht (2002)MATHGoogle Scholar
  23. 23.
    Klatte, D., Kummer, B.: Aubin property and uniqueness of solutions in cone constrained optimization. Math. Methods Oper. Res. 77, 291–304 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Levy, A.B.: Implicit multifunction theorems for the sensitivity analysis of variational conditions. Math. Program. 74, 333–350 (1996)MathSciNetMATHGoogle Scholar
  25. 25.
    Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mordukhovich, B.S.: Complete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–35 (1993)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  28. 28.
    Mordukhovich, B.S.: Comments on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it. TOP 23, 35–42 (2015)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Mordukhovich, B.S., Nghia, T.T.A.: Full Lipschitzian and Holderian stability in optimization with applications to mathematical programming and optimal control. SIAM J. Optim. 24, 1344–1381 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Mordukhovich, B.S., Nghia, T.T.A.: Second-order characterizations of tilt stability with applications to nonlinear programming. Math. Program. 149, 83–104 (2015)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mordukhovich, B.S., Nghia, T.T.A.: Local monotonicity and full stability for parametric variational systems. SIAM J. Optim. 26, 1032–1059 (2016)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Mordukhovich, B.S., Nghia, T.T.A., Rockafellar, R.T.: Full stability in finite-dimensional optimization. Math. Oper. Res. 40, 226–252 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Mordukhovich, B.S., Outrata, J.V., Ramírez, C.H.: Second-order variational analysis in conic programming with applications to optimality and stability. SIAM J. Optim. 25, 76–101 (2015)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Mordukhovich, B.S., Outrata, J.V., Ramírez, C.H.: Graphical derivatives and stability analysis for parameterized equilibria with conic constraints. Set-Valued Var. Anal. 23, 687–704 (2015)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mordukhovich, B.S., Sarabi, M.E.: Variational analysis and full stability of optimal solutions to constrained and minimax problems. Nonlinear Anal. 121, 36–53 (2015)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Mordukhovich, B.S., Sarabi, M.E.: Generalized differentiation of piecewise linear functions in second-order variational analysis. Nonlinear Anal. 132, 240–273 (2016)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mordukhovich, B.S., Sarabi, M.E.: Second-order analysis of piecewise linear functions with applications to optimization and stability. J. Optim. Theory Appl. 171, 504–526 (2016)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Mordukhovich, B.S., Sarabi, M.E.: Stability analysis for composite optimization problems and parametric variational systems, to appear in. J. Optim. Theory Appl. 172, 554–577 (2017)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Pang, J.-S.: Convergence of splitting and Newton methods for complementarity problems: an application of some sensitivity results. Math. Program. 58, 149–160 (1993)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Robinson, S.M.: Generalized equations and their solutions, Part I: basic theory. Math. Program. Stud. 10, 128–141 (1979)CrossRefMATHGoogle Scholar
  43. 43.
    Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Stud. 14, 206–214 (1981)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Rockafellar, R.T.: First- and second-order epi-differentiability in nonlinear programming. Trans. Am. Math. Soc. 307, 75–108 (1988)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  46. 46.
    Rockafellar, R.T., Zagrodny, D.: A derivative-coderivative inclusion in second-order nonsmooth analysis. Set-Valued Anal. 5, 1–17 (1997)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.RUDN UniversityMoscowRussia
  3. 3.Department of MathematicsMiami UniversityOxfordUSA

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