Abstract
In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting function, an augmented Lagrange multiplier exists if and only if this penalty function is exact. We also develop a new general approach to the study of augmented Lagrange multipliers called the localization principle. The localization principle allows one to study the local behaviour of the augmented Lagrangian near global optimal solutions of the initial optimization problem in order to prove the existence of augmented Lagrange multipliers.
Similar content being viewed by others
References
Andreani, R., Birgin, E.G., Martinez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)
Bai, F.S., Wu, Z.Y., Zhu, D.L.: Lower order calmness and exact penalty function. Optim. Methods Softw. 21, 515–525 (2006)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New Jersey (2006)
Birgin, E.G., Floudas, C.A., Martinez, J.M.: Global minimization using an augmented Lagrangian method with variable lower-level constraints. Math. Prog. Ser. A. 125, 139–162 (2010)
Birgin, E.G., Martinez, J.M., Prudente, L.F.: Optimality properties of an augmented Lagrangian method on infeasible problems. Comput. Optim. Appl. 60, 609–631 (2015)
Burachik, R.S., Gasimov, R.N., Ismayilova, N.A., Kaya, C.Y.: On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J. Glob. Optim. 34, 55–78 (2006)
Burachik, R.S., Iusem, A.N., Melo, J.G.: Duality and exact penalization for general augmented Lagrangians. J. Optim. Theory Appl. 147, 125–140 (2010)
Burachik, R.S., Iusem, A.N., Melo, J.G.: The exact penalty map for nonsmooth and nonconvex optimization. Optimization 64, 717–738 (2015)
Burachik, R.S., Rubinov, A.: Abstract convexity and augmented Lagrangians. SIAM J. Optim. 18, 413–436 (2007)
Chatzipanagiotis, N., Dentcheva, D., Zavlanos, M.M.: An augmented Lagrangian method for distributed optimization. Math. Prog. Ser. A. 152, 405–434 (2015)
Curtis, F.E., Jiang, H., Robinson, D.P.: An adaptive augmented Lagrangian method for large-scale constrained optimization. Math. Prog. Ser. A. 152, 201–245 (2015)
Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization 65, 1167–1202 (2015)
Flores-Bazán, F., Mastroeni, G.: Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications. SIAM J. Optim. 25, 647–676 (2015)
Gasimov, R.N.: Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J. Glob. Optim. 24, 187–203 (2002)
Gasimov, R.N., Rubinov, A.M.: On augmented Lagrangian for optimization problems with a single constraint. J. Glob. Optim. 28, 153–173 (2004)
Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)
Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003)
Huang, X.X., Yang, X.Q.: Further study on augmented Lagrangian duality theory. J. Glob. Optim. 31, 193–210 (2005)
Hiriart-Urruty, J.-B., Strodiot, J.-J., Hien Nguyen, V.: Generalized Hessian matrix and second-order optimality conditions for problems with \(C^{1, 1}\) data. Appl. Math. Optim. 11, 43–56 (1984)
Kan, C., Song, W.: Augmented Lagrangian duality for composite optimization problems. J. Optim. Theory Appl. 165, 763–784 (2015)
Kan, C., Song, W.: Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems. J. Glob. Optim. 63, 77–97 (2015)
Kelley, J.L.: General Topology. Springer, New York (1975)
Klatte, D., Tammer, K.: On second-order sufficient optimality conditions for \(C^{1, 1}\)-optimization problems. Optimization 19, 169–179 (1988)
Liu, Q., Tang, W.M., Yang, X.M.: Properties of saddle points for generalized augmented Lagrangian. Math. Methods Oper. Res. 69, 111–124 (2009)
Liu, Q., Yang, X.: Zero duality and saddle points of a class of augmented Lagrangian functions in constrained non-convex optimization. Optimization 57, 655–667 (2008)
Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)
Meng, K.W., Yang, X.Q.: Optimality conditions via exact penalty functions. SIAM J. Optim. 20, 3208–3231 (2010)
Penot, J.-P.: Augmented Lagrangians, duality and growth conditions. J. Nonlinear Convex Anal. 3, 283–302 (2002)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, London (1969)
Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control 12, 268–285 (1974)
Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)
Rückmann, J.J., Shapiro, A.: Augmented Lagrangians in semi-infinite programming. Math. Prog. Ser. B. 116, 499–512 (2009)
Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Prog. Ser. B. 77, 301–320 (1997)
Shapiro, A., Sun, J.: Some properties of the augmented Lagrangian in cone constrained optimization. Math. Oper. Res. 29, 479–491 (2004)
Sun, X.L., Li, D., Mckinnon, K.: On saddle points of augmented Lagrangians for constrained nonconvex optimization. SIAM J. Optim. 15, 1128–1146 (2005)
Wang, C., Liu, Q., Qu, B.: Global saddle points of nonlinear augmented Lagrangian functions. J. Glob. Optim. (2016). doi:10.1007/s10898-016-0456-y
Wang, C.Y., Yang, X.Q., Yang, X.M.: Nonlinear augmented Lagrangian and duality theory. Math. Oper. Res. 38, 740–760 (2013)
Wang, C., Zhou, J., Xu, X.: Saddle points theory of two classes of augmented Lagrangians and its applications to generalized semi-infinite programming. Appl. Math. Optim. 59, 413–434 (2009)
Wu, Z.Y., Bai, F.S., Yang, X.Q., Zhang, L.S.: An exact lower order penalty function and its smoothing in nonlinear programming. Optimization 53, 51–68 (2004)
Wu, H.X., Luo, H.Z., Yang, J.F.: Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming. J. Glob. Optim. 59, 695–727 (2014)
Yang, X.Q., Meng, Z.Q.: Lagrange multipliers and calmness conditions of order \(p\). Math. Oper. Res. 32, 95–101 (2007)
Zhou, J., Chen, J.-S.: On the existence of saddle points for nonlinear second-order cone programming problems. J. Glob. Optim. 62, 459–480 (2015)
Zhou, J., Xiu, N., Wang, C.: Saddle point and exact penalty representation for generalized proximal Lagrangians. J. Glob. Optim. 56, 669–687 (2012)
Zhou, Y.Y., Yang, X.Q.: Some results about duality and exact penalization. J. Glob. Optim. 29, 497–509 (2004)
Zhou, Y.Y., Yang, X.Q.: Augmented Lagrangian function, non-quadratic growth condition and exact penalization. Oper. Res. Lett. 34, 127–134 (2006)
Zhang, L., Yang, X.: An augmented Lagrangian approach with a variable transformation in nonlinear programming. Nonlinear Anal. 69, 2095–2113 (2008)
Zhou, Y.Y., Yang, X.Q.: Duality and penalization in optimization via an augmented Lagrangian function with applications. J. Optim. Theory Appl. 140, 171–188 (2009)
Zhou, Y.Y., Yang, X.Q.: Augmented Lagrangian functions for constrained optimization problems. J. Glob. Optim. 52, 95–108 (2012)
Zhou, Y.Y., Zhou, J.C., Yang, X.Q.: Existence of augmented Lagrange multipliers for cone constrained optimization problems. J. Glob. Optim. 58, 243–260 (2014)
Acknowledgements
The author is grateful to the anonymous referees for thoughtful and stimulating comments that helped to improve the quality of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
The reported study was supported by Russian Foundation for Basic Research, research Project No. 16–31–00056.
Rights and permissions
About this article
Cite this article
Dolgopolik, M.V. Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle. Math. Program. 166, 297–326 (2017). https://doi.org/10.1007/s10107-017-1122-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-017-1122-y
Keywords
- Augmented Lagrangian
- Lagrange multipliers
- Exact penalty function
- Localization principle
- Nonlinear semidefinite programming