The Modified Inertial Proximal DC Algorithm

With the development and advancement of technology,especially the arrival of the era of big data,optimization theory and algorithm play an increasingly important role in practical applications.However,the mathematical model abstracted from most practical problems in real life is a non-convex optimization problem,which promotes the development and research of non-convex optimization theory and algorithm.In fact,most of the non-convex optimization problems can be equivalently transformed into DC optimization problems.Those problems are well known because the objective function is the DC function,which is the difference of two convex functions.As a class of non-convex optimization problems with a special structure,they have a wide application background in many practical problems,such as compressive sensing,subset selection in regression analysis,feature selection in Support Vector Machines,the sparse eigenvalue problem and so on.Since the DC optimization problems were proposed,they have attracted the attention of experts and scholars for more than 30 years.There are numerous algo-rithms to solve DC optimization problems,and the most classical one is DC algorithm.In this paper,we consider a class of optimization problems whose objective function is the sum of a smooth non-convex loss function with Lipschitz gradient and a DC function,they can be equivalently transformed into DC optimization problems.To solve these problems,a new algorithm was proposed based on the existing algorithm called modified inertial proximal DC algorithm.Specifically,the new algorithm incorporates two different extrapolation with pre-cious information into the the backward proximal step and forward gradient step in subproblem of proximal DC algorithm.Under the general parameter constraints,we prove that each limit point of the sequence generated by the new algorithm is a stationary point.Furthermore,under the condition of Kurdyka-Lojasiewicz(KL)inequality,the new algorithm is proved to be glob-ally convergent.Preliminary numerical experiments show that the advantages over the previous algorithms,and the results are compatible with the theoretical analysis.This paper consists of five chapters:the first chapter describes the background of the prob-lems,the domestic and foreign research status and the frame structure of the full text.The second chapter gives some preliminary knowledge needed for this article,preparing the way for the fol-lowing content.The constant parameters and variable parameters forms of the new algorithm are proposed in the third chapter and we analyzed the relationship between the new algorithm and the existing algorithm.The fourth chapter analyzes the convergence of the algorithm.The fifth chapter carries out numerical experiments,analyzes the numerical experiment results and summarizes the full text.