A Second Order Primal-Dual Method for Nonsmooth Convex Composite Optimization

We develop a second order primal-dual method for optimization problems in which the objective function is given by sum of strongly convex twice differentiable term and possibly nondifferentiable regularizer. After introducing an auxiliary variable, we utilize proximal operator nonsmooth regularizer to transform associated augmented Lagrangian into that once, but not twice, continuously differentiable. The saddle point this corresponds solution original problem. employ generalization Hessian define second-order updates on prove global exponential stability corresponding differential inclusion. Furthermore, globally convergent customized algorithm utilizes as merit function. show search direction can be computed efficiently quadratic/superlinear asymptotic convergence. use $\ell _1$ -regularized model predictive control problem designing distributed controller spatially invariant system demonstrate merits effectiveness our method.