Dualization and Automatic Distributed Parameter Selection of Total Generalized Variation via Bilevel Optimization

An efficient augmented Lagrangian method with semismooth Newton solver for total generalized variation

“…There are first-order optimization methods. To the best of our knowledge, the second-order semismooth Newton method is first discussed in [22] with additional Tikhonov regularization on the dual variables.…”

“…The proposed algorithm is based on applying ALM to the perturbed primal problem of TGV, where we can benefit from the strong convexity. With the ALM framework, we do not need the Tikhonov regularization on the dual variables as did in [22] for TGV regularized image restoration, where semismooth Newton method is applied directly to the corresponding optimality conditions. The ALM can be seen as a kind of globalization of semismooth Newton methods, which directly aims at the perturbed original problem without Tikhonov regularizations on the dual variables.…”

“…Semismooth Newton methods [25] can be employed directly to solve such kind of nonlinear equation (17). However, we will introduce another equivalent nonlinear system compared to (17) through new auxiliary variables, which is more convenient for semismooth Newton solvers as shown for some variants of TV model [23,41] and TGV model [22]. Actually, by the Moreau's equality [2,8] x…”

“…Henceforth, we will focus on semismooth Newton methods for solving (22) instead of (17). Our motivation is that the primal-dual semismooth Newton method to be discussed can be highly efficient for the nonlinear system (22) compared to semismooth Newton method for solving (17), which is also shown in [41] for TV regularization problems.…”

“…Semismooth Newton method and Newton derivative. In this section, we will look closely at the primal-dual semismooth Newton method and its delicate application to (22) along with the ALM for the primal problem (P). For semismooth Newton methods, the Newton derivative is of critical importance.…”