Lagrangian-Dual Functions and Moreau--Yosida Regularization

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 . This property is then used to investigate the second-order properties of the Moreau-Yosida regularization of the function , e.g., the semismoothness of the gradient g of the regularized function . We show that 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 Lagrangiandual function 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. Key words. Lagrangian dual, Mo...

• Type: Journal
• Year: 2008
• Where: SIAMJO
• Authors: Fanwen Meng, Gongyun Zhao, Mark Goh, Robert de Souza
• Keywords: Lagrangian Dual Problem, Optimization Problem, Regularized Function, SIAMJO 2008