A Generalized Balanced Customized Proximity Point Algorithm

With the development of science and technology,problems in fields such as engineering technology,signal processing,machine learning,automatic control systems,and economics and finance can all be transformed into optimization problems.Linear equality constrained convex optimization problem is one of the common optimization problems.In order to solve this problem,domestic and foreign scholars have provided many algorithms.When solving convex optimization problems with linear equality constraints,the Customized Proximity Point Algorithm(CPPA)is very simple and efficient.However,the algorithm requires strict parameter conditions and includes relaxation steps,which limits its applicability.In many practical applications,the relaxation step of the algorithm is not allowed.A Generalized Customized Proximity Point Algorithm(GCPPA)eliminates the relaxation step on the basis of the CPPA,while ensuring the convergence speed,relaxing the parameter conditions.Balanced Augmented Lagrange Method(B-ALM)is also a generalization of CPPA,which balances the computational difficulty of two subproblems and loosens the parameter conditions of the algorithm.A new Penalty ALM(P-ALM)algorithm further loosens the parameter conditions of B-ALM.In order to expand the applicability of the algorithm in practical applications,this paper proposes a generalized Balanced Customized Proximity Point Algorithm(BG-CPPA)based on the GCPPA and P-ALM.The new algorithm ensures the convergence speed and the parameter conditions are more relaxed,in the framework of variational inequalities,the convergence and convergence rate of the algorithm are proved.In the last numerical experiment of this paper,by solving the compression sensing problem and correlation matrix correction problem,the two problems are set up with different experimental accuracy and scale,which proves that the new algorithm is effective compared with the existing algorithm.