Extended alternating structure-adapted proximal gradient algorithm for nonconvex nonsmooth problems

TL;DR

This paper introduces an extended ASAP algorithm capable of efficiently solving complex nonconvex nonsmooth optimization problems with multiblock nonseparable structures, supported by convergence analysis and numerical validation.

Contribution

The paper develops an extended ASAP (eASAP) algorithm that handles multiblock nonseparable coupling functions in nonconvex nonsmooth optimization, with proven convergence properties.

Findings

eASAP effectively minimizes multiblock nonseparable objectives.

The algorithm exhibits global convergence under mild conditions.

Numerical results show superior performance in multimodal data fusion tasks.

Abstract

Alternating structure-adapted proximal (ASAP) gradient algorithm (M. Nikolova and P. Tan, SIAM J Optim, 29:2053-2078, 2019) has drawn much attention due to its efficiency in solving nonconvex nonsmooth optimization problems. However, the multiblock nonseparable structure confines the performance of ASAP to far-reaching practical problems, e.g., coupled tensor decomposition. In this paper, we propose an extended ASAP (eASAP) algorithm for nonconvex nonsmooth optimization whose objective is the sum of two nonseperable functions and a coupling one. By exploiting the blockwise restricted prox-regularity, eASAP is capable of minimizing the objective whose coupling function is multiblock nonseparable. Moreover, we analyze the global convergence of eASAP by virtue of the Aubin property on partial subdifferential mapping and the Kurdyka-{\L}ojasiewicz property on the objective. Furthermore, the sublinear convergence rate of eASAP is built upon the proximal point algorithmic framework under some mild conditions. Numerical simulations on multimodal data fusion demonstrate the compelling performance of the proposed method.