Abstract
We consider a problem of minimizing an extended real-valued function defined in a Hausdorff topological space. We study the dual problem induced by a general augmented Lagrangian function. Under a simple set of assumptions on this general augmented Lagrangian function, we obtain strong duality and existence of exact penalty parameter via an abstract convexity approach. We show that every cluster point of a sub-optimal path related to the dual problem is a primal solution. Our assumptions are more general than those recently considered in the related literature.
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Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003)
Nedić, A., Ozdaglar, A.: A geometric framework for nonconvex optimization duality using augmented Lagrangian functions. J. Global Optim. 40, 545–573 (2008)
Burachik, R.S., Rubinov, A.M.: Abstract convexity and augmented Lagrangians. SIAM J. Optim. 18, 413–436 (2007)
Rubinov, A.M., Huang, X.X., Yang, X.Q.: The zero duality gap property and lower semicontinuity of the perturbation function. Math. Oper. Res. 27, 775–791 (2002)
Zhou, Y.Y., Yang, X.Q.: Augmented Lagrangian function, non-quadratic growth condition and exact penalization. Oper. Res. Lett. 34, 127–134 (2006)
Wang, C.Y., Yang, X.Q., Yang, X.M.: Unified nonlinear Lagrangian approach to duality and optimal paths. J. Optim. Theory Appl. 135, 85–100 (2007)
Penot, J.P., Rubinov, A.M.: Multipliers and general Lagrangians. Optimization 54, 443–467 (2005)
Nedić, A., Ozdaglar, A., Rubinov, A.M.: Abstract convexity for nonconvex optimization duality. Optimization 56, 655–674 (2007)
Rubinov, A.M., Yang, X.Q.: Lagrange-type Functions in Constrained Non-convex Optimization. Kluwer Academic, Dordrecht (2003)
Burachik, R.S., Rubinov, A.M.: On the absence of duality gap for Lagrange-type functions. J. Ind. Manage. Optim. 1, 33–38 (2005)
Oettli, W., Yang, X.Q.: Modified Lagrangian and least root approaches for general nonlinear optimization problems. Acta Math. Appl. Sin. Engl. Ser. 18, 147–152 (2002)
Wang, C.Y., Yang, X.Q., Yang, X.M.: Nonlinear Lagrange duality theorems and penalty function methods in continuous optimization. J. Global Optim. 27, 473–484 (2003)
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)
Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer Academic, Dordrecht (2000)
Wang, C.Y., Yang, X.Q., Yang, X.M.: Unified nonlinear Lagrangian approach to duality and optimal paths. J. Optim. Theory Appl. 135(1), 85–100 (2007)
Zhang, L., Yang, X.Q.: An augmented Lagrangian approach with a variable transformation in nonlinear programming. Nonlinear Anal. 69, 2095–2113 (2008)
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Communicated by X.Q. Yang.
The authors are very grateful to the referees for their careful reading and correction of the previous version of the manuscript. In particular, the example in Remark 3.2 was motivated by an example provided by one of the referees. J.G. Melo was supported by CNPq-Brazil. J.G. Melo would like to thank the School of Mathematics and Statistics at the University of South Australia, for providing excellent conditions and a stimulating environment for carrying out his research.
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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). https://doi.org/10.1007/s10957-010-9711-4
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DOI: https://doi.org/10.1007/s10957-010-9711-4