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Title: Lagrangian-dual functions and Moreau-Yosida regularization
Authors: Meng, F.
Zhao, G. 
Goh, M. 
De Souza, R.
Keywords: Fenchel conjugate
Lagrangian dual
Moreau-Yosida regularization
Piecewise Ck functions
Issue Date: 2008
Citation: Meng, F., Zhao, G., Goh, M., De Souza, R. (2008). Lagrangian-dual functions and Moreau-Yosida regularization. SIAM Journal on Optimization 19 (1) : 39-61. ScholarBank@NUS Repository.
Abstract: In this paper, we consider the Lagrangian-dual problem of a class of convex optimization problems. We first discuss the semismoothness of the Lagrangian-dual function φ. This property is then used to investigate the second-order properties of the Moreau-Yosida regularization η of the function φ, e.g., the semismoothness of the gradient g of the regularized function η. We show that φ and g are piecewise C2 and semismooth, respectively, for certain instances of the optimization problem. We establish a relationship between the original problem and the Fenchel conjugate of the regularization of the corresponding Lagrangian dual problem. We also find some instances of the optimization problem whose Lagrangian-dual function φ is not piecewise smooth. However, its regularized function still possesses nice second-order properties. Finally, we provide an alternative way to study the semismoothness of the gradient under the structure of the epigraph of the dual function. © 2008 Society for Industrial and Applied Mathematics.
Source Title: SIAM Journal on Optimization
ISSN: 10526234
DOI: 10.1137/060673746
Appears in Collections:Staff Publications

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