Abstract
Given an algorithm A for solving some mathematical problem based on the iterative solution of simpler subproblems, an outer trust-region (OTR) modification of A is the result of adding a trust-region constraint to each subproblem. The trust-region size is adaptively updated according to the behavior of crucial variables. The new subproblems should not be more complex than the original ones, and the convergence properties of the OTR algorithm should be the same as those of Algorithm A. In the present work, the OTR approach is exploited in connection with the “greediness phenomenon” of nonlinear programming. Convergence results for an OTR version of an augmented Lagrangian method for nonconvex constrained optimization are proved, and numerical experiments are presented.
Similar content being viewed by others
References
Castelani, E.V., Martinez, A.L., Martínez, J.M., Svaiter, B.F.: Addressing the greediness phenomenon in nonlinear programming by means of proximal augmented Lagrangians. Comput. Optim. Appl. 46, 229–245 (2010)
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
Birgin, E.G., Castelani, E.V., Martinez, A.L., Martínez, J.M.: Outer trust-region method for constrained optimization. Technical Report MCDO251110, Department of Applied Mathematics, State University of Campinas (2010)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)
Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)
Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285 (1974)
Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On Augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)
Birgin, E.G., Floudas, C.A., Martínez, J.M.: Global minimization using an Augmented Lagrangian method with variable lower-level constraints. Math. Program. 125, 139–162 (2010)
Birgin, E.G., Martínez, J.M.: Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput. Optim. Appl. 23, 101–125 (2002)
Martínez, J.M., Santos, L.T.: Some new theoretical results on recursive quadratic programming algorithms. J. Optim. Theory Appl. 97, 435–454 (1998)
Birgin, E.G., Martínez, J.M.: Improving ultimate convergence of an Augmented Lagrangian method. Optim. Methods Softw. 23, 177–195 (2008)
Gould, N.I.M., Orban, D., Toint, Ph.L.: CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29, 373–394 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Liqun Qi.
This work was supported by PRONEX-CNPq/FAPERJ (E-26/111.449/2010-APQ1), FAPESP (Grants 2006/53768-0 and 2005/57684-2), and CNPq.
Rights and permissions
About this article
Cite this article
Birgin, E.G., Castelani, E.V., Martinez, A.L.M. et al. Outer Trust-Region Method for Constrained Optimization. J Optim Theory Appl 150, 142–155 (2011). https://doi.org/10.1007/s10957-011-9815-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9815-5