A Momentum-based Acceleration Gradient Method For Solving Strictly Convex Quadratic Minimization Problem And Its Extension

Strictly convex quadratic minimization is an important unconstrained optimization problem,which is widely used in image processing,machine learning and other fields.It provides a simple framework for the study of optimization algorithms.In addition,solving constrained optimization problems can be transformed into solving a series of convex quadratic minimization problems.Therefore,it is of great theoretical and practical significance to study the acceleration algorithm of strictly convex quadratic minimization.In this dissertation,we study the momentum-based acceleration gradient method for solving strictly convex quadratic problems and its extension.The main contents are as follows:Firstly,a momentum-based acceleration gradient method for strictly convex quadratic minimization is proposed.The main feature of this method is to introduce a new momentum term on the basis of auxiliary points,this momentum parameter minimizes the gradient norm of the objective function at the new iteration point and its step size minimizes the gradient norm of the objective function at the auxiliary point.Under certain conditions,the global convergence and -linear convergence rate of the method are proved.In addition,a momentum-based mixed acceleration gradient method is proposed to solve the strictly convex quadratic minimization problem.This method is a generalization of momentum-based acceleration gradient method.The step size is the convex combination of Cauchy step size and minimal gradient steplength.This method is not only suitable for the strictly convex quadratic minimization problem with small condition number,but also has good numerical performance for the problem with large condition number.Under suitable conditions,the global convergence and -linear convergence rate of this algorithm are discussed.Numerical results show that these two algorithms are effective compared with classical algorithms.Secondly,a momentum-based acceleration gradient method is proposed for solving strongly convex minimization problem.It is a generalization of momentum-based acceleration gradient method for solving strictly convex quadratic minimization problem.The step size of the algorithm is obtained by Armijo line search in the sense of gradient norm,and a new iteration point is generated by the accept-reject strategy.Under mild conditions,the global convergence and -linear convergence rate of the algorithm are analyzed.When the parameter introduced in the algorithm takes a value in a certain range,the algorithm does not need to implement the Armijo line search strategy in the sense of the gradient norm.When the objective function is strictly convex quadratic minimization problem and the introduced parameter =1,the algorithm becomes a momentum-based acceleration gradient method for solving strictly convex quadratic minimization problem.Numerical experiments show that the proposed algorithm is effective compared with existing ones.