Skip to main content
Log in

Augmented Lagrangians and hidden convexity in sufficient conditions for local optimality

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

Second-order sufficient conditions for local optimality have long been central to designing solution algorithms and justifying claims about their convergence. Here a far-reaching extension of such conditions, called variational sufficiency, is explored in territory beyond just classical nonlinear programming. Variational sufficiency is already known to support multiplier methods that are able, even without convexity, to achieve problem decomposition, but further insight has been needed into how it coordinates with other sufficient conditions. In the framework of this paper, it is shown to characterize local optimality in terms of having a convex–concave-type local saddle point of an augmented Lagrangian function. A stronger version of variational sufficiency is tied in turn to local strong convexity in the primal argument of that function and a property of augmented tilt stability that offers crucial aid to Lagrange multiplier methods at a fundamental level of analysis. Moreover, that strong version is translated here through second-order variational analysis into statements that can readily be compared to existing sufficient conditions in nonlinear programming, second-order cone programming, and other problem formulations which can incorporate nonsmooth objectives and regularization terms.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Aademic Press (1982)

  3. Bonnans, J.F., Ramirez, H.C.: Perturbation analysis of second-order cone programming problems. Math. Program. 104, 205–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dontchev, A.D., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View From Variational Analysis, Springer, 2009; second edition (2014)

  5. Hang, N.V.T., Mordukhovich, B.S., Sarabi, M.E.: Second-order variational analysis in second-order cone programming. Math. Program. 180, 930–946 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hang, N.V.T., Mordukhovich, B.S., Sarabi, M.E.: Augmented Lagrangian method for second-order cone programs under second-order sufficiency. J. Global Optim. (2021). https://doi.org/10.1007/s10898-021-0168-1

    Article  MATH  Google Scholar 

  7. Mohammadi, A., Mordukhovich, B.S., Sarabi, M.E.: Parabolic regularity in geometric variational analysis. Trans. Am. Math. Soc. 374, 1711–1763 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, SIAM Volumes in Applied Mathematics, vol. 58, pp. 32–46 (1992)

  9. Mordukhovich, B.S.: Lipschitzian stability of constraint systems and generalized equations. Nonlinear Anal. Theory Methods Appl. 22, 173–206 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mordukhovich, B.S., Outrata, J.V., Sarabi, M.E.: Full stability of locally optimal solutions in second-order cone programming. SIAM J. Optim. 24, 1581–1613 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 170–191 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Poliquin, R.A.: Subgradient monotonicity and convex functions. Nonlinear Anal. Theory Methods Appl. 14, 385–398 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–289 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)

  15. Rockafellar, R.T.: Conjugate Duality and Optimization. In: No. 16 in Conference Board of Math\(\dot{{\rm S}}\)ciences Series, SIAM Publications (1974)

  16. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rockafellar, R.T.: Nonlinear optimization and related topics. In: di Pillo, G., Giannessi, F. (eds.) Extended Nonlinear Programming, pp. 381–399. Kluwer (1999)

  19. Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rockafellar, R.T.: Variational convexity and local monotonicity of subgradient mappings. Vietnam J. Math. 47, 547–561 (2019). https://doi.org/10.1007/s10013-019-00339-5

    Article  MathSciNet  MATH  Google Scholar 

  21. Rockafellar, R.T.: Progressive decoupling of linkages in optimization and variational inequalities with elicitable convexity or monotonicity. Set-Valued Var. Anal. 27, 863–893 (2019). https://doi.org/10.1007/s1128-018-0496-1

    Article  MathSciNet  MATH  Google Scholar 

  22. Rockafellar, R.T.: Convergence of augmented Lagrangian methods in extensions beyond nonlinear programming. (forthcoming)

  23. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. In: No. 317 in the series Grundlehren der Mathematischen Wissenschaften, Springer (1998)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Tyrrell Rockafellar.

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

Rockafellar, R.T. Augmented Lagrangians and hidden convexity in sufficient conditions for local optimality. Math. Program. 198, 159–194 (2023). https://doi.org/10.1007/s10107-022-01768-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-022-01768-w

Keywords

Mathematics Subject Classification

Navigation