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 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 C 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 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.