In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth c...In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function.We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic firstand second-order oracles.The approach combines stochastic semismooth Newton steps,stochastic proximal gradient steps and a globalization strategy based on growth conditions.We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates.Under standard local assumptions,we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability.展开更多
This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of ...This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of Signorini type. We show that after a suitable Moreau-Yosida type regularization of the problem superlinear local convergence is obtained for a class of semismooth Newton methods. In addition, estimates for the order of tile error introduced by the regularization are derived. The main part of the paper is devoted to the analysis of a multilevel preconditioner for the semismooth Newton system. We prove a rigorous bound for the contraction rate of the multigrid cycle which is robust with respect to sufficiently small regularization parameters and the number of grid levels. Moreover, it applies to adaptively refined grids. The paper concludes with numerical results.展开更多
基金supported by the Fundamental Research Fund—Shenzhen Research Institute for Big Data Startup Fund(Grant No.JCYJ-AM20190601)the Shenzhen Institute of Artificial Intelligence and Robotics for Society+2 种基金National Natural Science Foundation of China(Grant Nos.11831002 and 11871135)the Key-Area Research and Development Program of Guangdong Province(Grant No.2019B121204008)Beijing Academy of Artificial Intelligence。
文摘In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function.We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic firstand second-order oracles.The approach combines stochastic semismooth Newton steps,stochastic proximal gradient steps and a globalization strategy based on growth conditions.We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates.Under standard local assumptions,we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability.
文摘This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of Signorini type. We show that after a suitable Moreau-Yosida type regularization of the problem superlinear local convergence is obtained for a class of semismooth Newton methods. In addition, estimates for the order of tile error introduced by the regularization are derived. The main part of the paper is devoted to the analysis of a multilevel preconditioner for the semismooth Newton system. We prove a rigorous bound for the contraction rate of the multigrid cycle which is robust with respect to sufficiently small regularization parameters and the number of grid levels. Moreover, it applies to adaptively refined grids. The paper concludes with numerical results.
