Lagrangian-Dual Functions and Moreau–Yosida Regularization
Fanwen Meng, Gongyun Zhao, Mark Goh, Robert de Souza · SIAM Journal on Optimization · 2008
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 $\varphi$. This property is then used to investigate the second-order properties of the Moreau–Yosida regularization $\eta$ of the function $\varphi$, e.g., the semismoothness of the gradient g of the regularized function $\eta$. We show that $\varphi$ and g are piecewise $C^2$ 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 $\varphi$ 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.