On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization
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On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization Gulcin Dinc Yalcin1 · Refail Kasimbeyli1 Received: 29 May 2019 / Revised: 9 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, zero duality gap conditions in nonconvex optimization are investigated. It is considered that dual problems can be constructed with respect to the weak conjugate functions, and/or directly by using an augmented Lagrangian formulation. Both of these approaches and the related strong duality theorems are studied and compared in this paper. By using the weak conjugate functions approach, special cases related to the optimization problems with equality and inequality constraints are studied and the zero duality gap conditions in terms of objective and constraint functions, are established. Illustrative examples are provided. Keywords Weak conjugacy · Augmented Lagrangians · Weak subdifferential · Nonconvex analysis · Duality Mathematics Subject Classification 90C26 · 90C30 · 90C46
1 Introduction Duality plays a very important role for establishing optimality conditions and developing solution methods in optimization. However, the dual problems developed for these purposes, provide efficient tools, if the corresponding zero duality gap relations are satisfied. In convex analysis, dual problems constructed in terms of conjugate functions by using the well-known Legendre-Fenchel transform, lead to the conventional Lagrangian function, and provide strong duality relations (Rockafellar 1974).
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Refail Kasimbeyli [email protected] Gulcin Dinc Yalcin [email protected]
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Department of Industrial Engineering, Faculty of Engineering, Eskisehir Technical University, Iki Eylul Campus, 26555 Eskisehir, Turkey
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G. D. Yalcin, R. Kasimbeyli
Nevertheless, for an optimization problem, without convexity conditions, the conventional Lagrangian function may not always guarantee the zero duality gap which may occur between the given (primal) problem and the dual one. On the other hand, the additional assumptions may need to have zero duality gap when the linear Lagrangian is considered (see Ernst and Volle 2013; Goberna et al. 2014) for the convex and (Ernst and Volle 2016; Flores-Bazan et al. 2017; Jeyakumar et al. 2009; Polik and Terlaky 2007; Polyak 1998) for some nonconvex cases). To construct dual problems and establish zero duality gap relations for nonconvex optimization problems, various generalizations of conjugate functions and Lagrangian functions have been proposed in the literature. The convex conjugate framework of Rockafellar (1974) was extended in Azimov and Gasimov (1999, 2002), Balder (1977), Gasimov (1992), Ioffe (1979) and Sharikov (2009) to nonconvex optimization. Duality theory which uses nonlinear Lagrangian functions, based on the notion of convolution functions, was investigated e.g. in Goh and Yang (2001), Li (1995), (Rubinov et al. 1999a, b, 2002, 2003), Wang et al. (2017) and Yang and Huang (2001), where the zero duality gap properties we
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