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

• 成果类型: Article
• 署名作者: Dhingra, Neil K.; Khong, Sei Zhen; Jovanovic, Mihailo R.
• 署名单位: University of Southern California
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3115449
• 发表日期: 2022
• 页码: 4061-4076
• 关键词: Control design; Control system synthesis; design optimization; Distributed algorithms; gradient methods; Newton method; optimization; Optimization methods

摘要

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 secondorder 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 l(1) -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.