科研之友
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摘要
The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: (1) Many well-known operator splitting methods, such as forward-backward splitting and Douglas-Rachford splitting, actually define a fixed-point mapping; (2) The optimal solutions of the composite convex program and the solutions of a system of nonlinear equations derived from the fixed-point mapping are equivalent. Solving this kind of system of nonlinear equations enables us to develop second-order type methods. These nonlinear equations may be non-differentiable, but they are often semi-smooth and their generalized Jacobian matrix is positive semidefinite due to monotonicity. By combining with a regularization approach and a known hyperplane projection technique, we propose an adaptive semi-smooth Newton method and establish its convergence to global optimality. Preliminary numerical results on -minimization problems demonstrate that our second-order type algorithms are able to achieve superlinear or quadratic convergence.
