Approximate Bregman proximal gradient algorithm with variable metric Armijo--Wolfe line search

Abstract: We propose a variant of the approximate Bregman proximal gradient (ABPG) algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. Although ABPG is known to converge globally to a stationary point even when the smooth part of the objective function lacks globally Lipschitz continuous gradients, and its iterates can often be expressed in closed form, ABPG relies on an Armijo line search to guarantee global convergence. Such reliance can slow down performance in practice. To overcome this limitation, we propose the ABPG with a variable metric Armijo--Wolfe line search. Under the variable metric Armijo--Wolfe condition, we establish the global subsequential convergence of our algorithm. Moreover, assuming the Kurdyka--\L{}ojasiewicz property, we also establish that our algorithm globally converges to a stationary point. Numerical experiments on $\ell_p$ regularized least squares problems and nonnegative linear inverse problems demonstrate that our algorithm outperforms existing algorithms.