ABSTRACT
In this paper, we propose a framework of Inexact Proximal Stochastic Second-order (IPSS) method for solving nonconvex optimization problems, whose objective function consists of an average of finitely many, possibly weakly, smooth functions and a convex but possibly nonsmooth function. At each iteration, IPSS inexactly solves a proximal subproblem constructed by using some positive definite matrix which could capture the second-order information of original problem. Proper tolerances are given for the subproblem solution in order to maintain global convergence and the desired overall complexity of the algorithm. Under mild conditions, we analyse the computational complexity related to the evaluations on the component gradient of the smooth function. We also investigate the number of evaluations of subgradient when using an iterative subgradient method to solve the subproblem. In addition, based on IPSS, we propose a linearly convergent algorithm under the proximal Polyak–Łojasiewicz condition. Finally, we extend the analysis to problems with weakly smooth function and obtain the computational complexity accordingly.
Acknowledgments
We would like to thank two anonymous referees for their valuable comments and suggestions.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1 means that
is continuously differentiable.
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Xiao Wang
Xiao Wang is an associate professor at School of Mathematical Sciences at University of Chinese Academy of Sciences, China. Her research focuses on optimization theory, algorithms and applications. Her latest work is published in Optimization Methods and Software, Mathematics of Computation, and SIAM Journal on Optimization.
Hongchao Zhang
Hongchao Zhang is an Associate Professor at the Department of Mathematics at Louisiana State University, USA. His research focuses on nonlinear optimization theory and its applications. His latest work is published at Computational Optimization and Applications, Journal of Scientific Computing and SIAM Journal on Optimization.