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 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.