A note on embeddings for the Augmented Lagrange Method
DOI:
https://doi.org/10.2298/YJOR1002183BKeywords:
Augmented Lagrangian Method, JJT-regular, generalized critical points, generic setAbstract
Nonlinear programs (P) can be solved by embedding problem P into one parametric problem P(t), where P(1) and P are equivalent and P(0), has an evident solution. Some embeddings fulfill that the solutions of the corresponding problem P(t) can be interpreted as the points computed by the Augmented Lagrange Method on P. In this paper we study the Augmented Lagrangian embedding proposed in [6]. Roughly speaking, we investigated the properties of the solutions of P(t) for generic nonlinear programs P with equality constraints and the characterization of P(t) for almost every quadratic perturbation on the objective function of P and linear on the functions defining the equality constraints.References
Avelino, C., and Vicente, L.N, “Updating the multipliers associated with inequality constraints in an augmented Lagrangian multiplier method”, Journal of Optimization Theory and Applications, 119 (2003) 215–233.
Bazaraa, M.S., Sherali, H.D., and Shetty, C.M., Non Linear Programming Theory and Algorithms, John Wiley and Sons, 1993.
Bertsekas, D.P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.
Bouza, G., “A new embedding for the augmented Lagrangean method”, Investigacion Operacional, 22 (3) (2001) 145–153.
Chen, B., and Chen, X., “A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints”, Computational Optimization, 17 (2000) 131–158.
Dentcheva, D., Gollmer, R., Guddat, J., and Ruckmann, J., “Pathfollowing methods in non linear optimization, multipliers embedding”, ZOR, 41 (1995) 127–152.
Dostal, Z., Friedlander, A., and Santos, A., “Augmented Lagrangians with adaptive precision control for quadratic programming problems with equality constraints”, Computer Optimization Applications, 14 (1) (1999) 37–53.
Gollmer, R., Kausmann, U., Nowack, D., Wendler, K., and Bacallao Estrada, J., Computer Program PAFO, Humboldt-Universität, Institut für Mathematik, 2004.
Gomez, W., “On generic quadratic penalty embeddings for non linear optimization problems”, Optimization, 50 (3-4) (2001) 279–295.
Gomez, W., Guddat, J., Jongen, H. Th., Ruckmann, J.J., and Solano, C., Curvas críticas y saltos en la optimización no lineal. http://www.emis.de/monographs/curvas/index.html, 2000.
Guddat, J., Guerra-Vasquez, F., and Jongen, H. Th., Parametric Optimization: Singularities, Pathfollowing and Jumps, Teubner and John Wiley, Chichester, 1990.
Hirsch, M., Differential Topology. Springer Verlag, New York, 1976.
Iusem, A.N., “Augmented Lagrangean methods and proximal point methods for convex optimization”, Investigacion Operativa, 8 (1999) 11–49.
Jongen, H.Th., Jonker, P., and Twilt, F., “Critical sets in parametric optimization”, Mathematical Programming, 34 (3) (1986) 333–353.
Jongen, H.Th., Jonker, P., and Twilt, F., “On one-parameter families of optimization problems: Equality constraints”, Journal of Optimization Theory and Applications, 48 (1) (1986) 141–161.
Li, D., and Sun, X.L., “Local convexification of the Lagrangian function in non-convex optimization”, Journal of Optimization Theory and Applications, 104 (1) (2000) 109–120.
Lin, Z., and Song, D., “A continuation method for solving convex programming via Fisher reformulation”, Optimization, 44 (3) (1998) 291–302.
Luenberger, D.G., Linear and Nonlinear Programming, Addison-Wesley Inc., Reading, Massachusetts, Second edition, 1984.
Schmidt, R., Eine Modifizierte Standard Einbettung zur Behandlung von Gleichungs und Ungleichungs Restriktionen, Master’s thesis, Humboldt Universität zu Berlin, 2000.
Song, X., “A non-interior pathfollowing method for convex quadratic programming problems with bound constraints”, Computational Optimization and Applications, 27 (3) (2004) 285–303.
Watson, L., “Theory of globally convergent probability-one homotopies for nonlinear programming”, SIAM Journal on Optimization, 11 (3) (2000) 761–780.
Wright, S.J., “A path-following interior point algorithm for linear and quadratic problems”, Annals of Operations Research, 62 (1994) 103–130.
Downloads
Published
Issue
Section
License
Copyright (c) 2010 YUJOR
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
