2017/09/05 by Dhingra, Neil K., Khong, Sei Zhen, Jovanović, Mihailo R. · 1 citation
#Adaptation and Self-Organizing Systems (nlin.AO) #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #FOS: Electrical engineering #FOS: Mathematics #FOS: Physical sciences #Optimization and Control (math.OC) #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · doi:10.48550/arxiv.1709.01610
We develop a second order primal-dual method for optimization problems in which the objective function is given by the sum of a strongly convex twice differentiable term and a possibly nondifferentiable convex regularizer. After introducing an auxiliary variable, we utilize the proximal operator of the nonsmooth regularizer to transform the associated augmented Lagrangian into a function that is once, but not twice, continuously differentiable. The saddle point of this function corresponds to the solution of the original optimization problem. We employ a generalization of the Hessian to define second order updates on this function and prove global exponential stability of the corresponding differential inclusion. Furthermore, we develop a globally convergent customized algorithm that utilizes the primal-dual augmented Lagrangian as a merit function. We show that the search direction can be computed efficiently and prove quadratic/superlinear asymptotic convergence. We use the ℓ1-regularized model predictive control problem and the problem of designing a distributed controller for a spatially-invariant system to demonstrate the merits and the effectiveness of our method.