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.
Highlights
We consider a special class of nonconvex and nonsmooth composite problem
The problem is constituted by the sum of two functions, one is finite convex with inexact information and the other is a nonconvex function
We regard the sum of the convex function and the augment function as an approximate function
Summary
Consider the following optimization problem: Machado min ψ( x ) := f ( x ) + h( x ), x ∈R N. The proximal alternating linearization type methods (see [4,27,28,29]) are one effective kind of bundle methods for some composite problems They need to solve two subproblems at each iteration and the data referred are usually exact. We design a proximal bundle method for the inexact composite problem (1). The design and convergence analysis of bundle methods for nonconvex problems with inexact function and subgradient evaluations are quite involved and there are only a handful of papers for this topic, see [15,32,33,34,35]. We present a proximal bundle method with a convexification technique and noise management strategy to solve composite problem (1).
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