Accelerated Bregman Proximal Difference-of-Convex Algorithm For Nonconvex Nonsmooth Optimization

In recent years,with the development of big data and artificial intelligence,more and more practical problems have been transformed into mathematical models.The research on nonconvex optimization problems is also widely carried out.One of the special nonconvex optimization problems is DC optimization problems,whose objective function can be written as the difference of two convex functions.DC optimization has been studied for more than 30 years,and its content continues to expand,making it widely used in machine learning,compressed sensing,financial optimization and other fields.In this thesis,we consider a class of nonconvex and nonsmooth optimization problems which objective function is the sum of a continuous differentiable convex function,a proper closed convex function and a continuous concave function.The problem can be expressed as the difference of two convex functions,so it is also a class of DC optimization problems.We propose an accelerated bregman proximal DC algorithm(BPDCAe),which extends the proximal DC algorithm with extrapolation by introducing bregman distance,and overcomes the restriction that the differentiable part of the objective function needs to satisfy the global gradient Lipschitz continuity.Based on the bregman distance,this paper introduces the smooth adaptive condition and gives the extended descent lemma.In the convergence analysis,it is proved that any accumulation point of the sequence generated by BPDCAe is a stationary point of the optimization problem under the appropriate parameter selection.In addition,under the Kurdyka-Lojasiewicz property,the global convergence of the algorithm is proved,that is,the sequence generated by BPDCAe converges to a stationary point of the optimization problem.Finally,in order to verify the feasibility and effectiveness of BPDCAe,BPDCAe is applied to the Poisson linear inverse problem.The numerical experimental results show that BPDCAe has fewer iterations,faster speed and is more effective than BPDCA.