Dunbiao Niu

Qihao Ma, Chengjing Wang, Peipei Tang et al.

This paper proposes a novel distributed semismooth Newton based augmented Lagrangian method for solving a class of optimization problems over networks, where the global objective is defined as the sum of locally held cost functions, and communication is restricted to neighboring agents. Specifically, we employ the augmented Lagrangian method to solve an equivalently reformulated constrained version of the original problem. Each resulting subproblem is solved inexactly via a distributed semismooth Newton method. By fully leveraging the structure of the generalized Hessian, a distributed accelerated proximal gradient method is proposed to compute the Newton direction efficiently, eliminating the need to communicate with full Hessian matrices. Theoretical results are also obtained to guarantee the convergence of the proposed algorithm. Numerical experiments demonstrate the efficiency and superiority of our algorithm compared to state-of-the-art distributed algorithms.

Dunbiao Niu, Chengjing Wang, Peipei Tang et al.

Support vector machines (SVMs) are successful modeling and prediction tools with a variety of applications. Previous work has demonstrated the superiority of the SVMs in dealing with the high dimensional, low sample size problems. However, the numerical difficulties of the SVMs will become severe with the increase of the sample size. Although there exist many solvers for the SVMs, only few of them are designed by exploiting the special structures of the SVMs. In this paper, we propose a highly efficient sparse semismooth Newton based augmented Lagrangian method for solving a large-scale convex quadratic programming problem with a linear equality constraint and a simple box constraint, which is generated from the dual problems of the SVMs. By leveraging the primal-dual error bound result, the fast local convergence rate of the augmented Lagrangian method can be guaranteed. Furthermore, by exploiting the second-order sparsity of the problem when using the semismooth Newton method,the algorithm can efficiently solve the aforementioned difficult problems. Finally, numerical comparisons demonstrate that the proposed algorithm outperforms the current state-of-the-art solvers for the large-scale SVMs.