  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.
