Research On Inexact Multilevel Gradient Mirror Descent Algorithm For Composite Convex Optimization Problem And Their Application

Composite convex optimization is an important type of optimization problem,usually expressed as the sum of smooth convex function and non-smooth convex function.this model is often used in artificial intelligence and machine learning,including compressed sensing,data classification,and signal and image processing.Due to the wide application of the composite convex optimization problem,This paper proposes two algorithms to solve composite convex optimization problems based on reasonable first-order method and second-order approximation.And the corresponding numerical experiments verify the effectiveness of the algorithms.The following is the main research work of this paper:Firstly,based on the idea of multi-level optimization,gradient mirror descent algorithm and ?-sub-differentiation,an inexact multi-level gradient mirror descent algorithm is proposed to solve the composite convex optimization problem.This algorithm allows errors in both the gradient calculation of smooth part and the proximal operator calculation of non-smooth part in the objective function.Under appropriate conditions,the O(1/k2)convergence speed of the function value error sequence of the algorithm is proved,where k represents the number of iterations.The application of this algorithm in Lasso problem,Logistic regression problem and image deblur is given.Finally,based on the proximal quasi-Newton algorithm and the idea of multi-level optimization,a multi-level proximal quasi-Newton algorithm for solving composite convex optimization problems is proposed.The application of this algorithm in logistic regression is given,The effectiveness of the algorithm is verified by numerical experiments.