An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization

Xiaoliang Wang, Liping Pang, Qi Wu and Mingkun Zhang
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Xiaoliang Wang: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Liping Pang: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Qi Wu: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Mingkun Zhang: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Mathematics, 2021, vol. 9, issue 8, 1-27

Abstract: In this paper, an adaptive proximal bundle method is proposed for a class of nonconvex and nonsmooth composite problems with inexact information. The composite problems are the sum of a finite convex function with inexact information and a nonconvex function. For the nonconvex function, we design the convexification technique and ensure the linearization errors of its augment function to be nonnegative. Then, the sum of the convex function and the augment function is regarded as an approximate function to the primal problem. For the approximate function, we adopt a disaggregate strategy and regard the sum of cutting plane models of the convex function and the augment function as a cutting plane model for the approximate function. Then, we give the adaptive nonconvex proximal bundle method. Meanwhile, for the convex function with inexact information, we utilize the noise management strategy and update the proximal parameter to reduce the influence of inexact information. The method can obtain an approximate solution. Two polynomial functions and six DC problems are referred to in the numerical experiment. The preliminary numerical results show that our algorithm is effective and reliable.

Keywords: nonconvex and nonsmooth; inexact oracles; disaggregate strategy; proximal bundle method; convexification technology; DC problems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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