A Second Order Primal-Dual Method for Nonsmooth Convex Composite Optimization
A Second Order Primal-Dual Method for Nonsmooth Convex Composite Optimization
复制标题
非光滑凸复合优化的二阶原对偶方法
DOI：
10.1109/tac.2021.3115449
复制
发表时间：
2017
影响因子：
6.8
通讯作者：
M. Jovanović
中科院分区：
文献类型：
--
作者：
Neil K. Dhingra;Sei Zhen Khong;M. Jovanović
我们开发了一个二阶原始-对偶方法的优化问题，其中的目标函数是由一个强凸二次可微项和一个可能不可微的凸正则化。在引入一个辅助变量后，我们利用非光滑正则化的近似算子将相关的增广拉格朗日变换成一个函数，该函数是一次，但不是两次，连续可微的。该函数的鞍点对应于原始优化问题的解。我们采用广义的Hessian定义二阶更新这个功能，并证明相应的微分包含的全局指数稳定性。此外，我们开发了一个全球收敛的定制算法，利用原始-对偶增广拉格朗日作为一个价值函数。我们证明了搜索方向可以有效地计算，并证明了二次/超线性渐近收敛。我们使用的$\ell_1 $-正则化模型预测控制问题和问题的设计一个分布式控制器的空间不变的系统，以证明我们的方法的优点和有效性。
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 $\ell _1$-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.
影响因子：
5.8
作者：
Friedman, Jerome;Hastie, Trevor;Tibshirani, Rob
通讯作者：
Tibshirani, Rob