Abstract
In this paper, we introduce the concept of the valley at 0 augmenting function and apply it to construct a class of valley at 0 augmented Lagrangian functions. We establish the existence of a path of optimal solutions generated by valley at 0 augmented Lagrangian problems and its convergence toward the optimal set of the original problem and obtain the zero duality gap property between the primal problem and the valley at 0 augmented Lagrangian dual problem. Moreover, we establish the exact penalization representation results in the framework of valley at 0 augmented Lagrangian.
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References
Auslender, A., Cominetti, R. and Haddou, M. (1997), Asymptotical analysis for penalty and barrier methods in convex and linear programming, Mathematics of Operations Research, 22, 43–62.
Bestsekas, D.P. (1982), Constrained Optimization and Lagrangian Multiplier Method, Academic Press, New York.
Huang, X.X. and Yang, X.Q. A unified augmented Lagrangian approach to duality and exact penalization, Mathematics of Operation Research, (accepted).
Ioffe, A. (1979), Necessary and sufficient conditions for a local minimum. 3: second-order conditions and augmented duality, SIAM Journal Control and Optimization, 17, 266–288.
Luo, Z.Q., Pang, J.S. and Ralph, D. (1996), Mathematical Programs with Equilibrium Constraints, Cambridge University Press, New York.
Rockafellar, R.T. (1974), Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM Journal on Control and Optimization, 12, 268–285.
Rockafellar, R.T. (1993), Lagrange multipliers and optimality, SIAM Review, 35, 183–238.
Rockafellar, R.T. and Wets, R.J.-B. (1998), Variational Analysis, Springer-Verlag, Berlin.
Rubinov, A.M., Huang, X.X. and Yang, X.Q. The zero duality gap property and lower semicontinuity of the perturbation function, Mathematics of Operation Research, (accepted).
Rubinov, A.M. and Yang, X.Q. (2003), Lagrange-type Functions in Constrained Non-convex Optimization, Kluwer Academic Publishers, Dordrecht.
Rubinov, A.M., Glover, B.M. and Yang, X.Q. (1999), Decreasing functions with applications to penalization, SIAM Journal of Optimization, 10(1), 289–313.
Yang, X.Q. and Huang, X.X. (2001), A nonlinear Lagrangian approach to constrained optimization problems, SIAM Journal of Optimization, 14, 1119–1144.
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Zhou, Y., Yang, X. Some Results about Duality and Exact Penalization. Journal of Global Optimization 29, 497–509 (2004). https://doi.org/10.1023/B:JOGO.0000047916.73871.88
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DOI: https://doi.org/10.1023/B:JOGO.0000047916.73871.88