A Kind Of Adaptive Generalized Alternating Direction Multiplier Method

Convex programming is a very common and important problem in the field of optimization.The applications of convex programming is not only in the s-tudy of mathematical problems,but also in the research of engineering science,management science and other disciplines.And it is widely used in mathemati-cal programming,image processing,traffic equilibrium and site selection,network economy and so on.In this paper,we consider the convex minimization problem with linear con-straints and an objective function in form of the sum of two functions without coupled variables.There are many algorithms to solve this kind of problem,how-ever,how to solve the problem more effectively still needs to be further explored.Among many algorithms,the alternating direction method of multipliers(ADM-M)have drawn the attention of scholars for its simplicity and effectiveness.As a result,many new algorithms have been developed,such as linearized linearized alternating direction method of multipliers(L-ADMM),generalized alternating di-rection method of multipliers(G-ADMM)and so on,and they are also widely used in various fields.Considering that when the subproblems are difficult to solve in the actual problems,the method of adding a proximal regularization term in the subproblem can be adopted.The convergence of the above algorithms is ensured when the proximal matrix is positive definite,and thus tiny step sizes inevitably occur.Some researches have shown that it is not necessary to ensure the positive definiteness of the proximal matrix.In this paper,we investigate the self-adaptive technique of the positive-indefinite proximal version of G-ADMM and report some preliminary numerical results.In chapter 2,we adopt the self-adaptive technique in the the positive-indefinite proximal version of G-ADMM[11],then dynamically select the proximal matrix with self-adaptive technique to increase step sizes.And we prove the global con-vergence of the proposed algorithm under some mild conditions.In chapter 3,we apply our algorithms to solve the LASSO and image restora-tion problems,and compare with other algorithms.The preliminary numerical results indicate that the new algorithm is feasible and efficient.