Research On Fast Iterative Shrinage-thresholding Algorithm For Non-smooth Convex Optimization Problems

The fast iterative shrinkage-thresholding algorithm?FISTA?is an effective algorithm for solving unconstrained optimization problems.Due to its advantages of easy implementation,low storage requirements,and good calculation effect,FISTA has attracted the attention of many scholars.FISTA has been generalized to constrained optimization and non-smooth optimization,and has been widely used in image processing and compressive sensing.Designing efficient and improved FISTA for different problems is one of the research hotspots in recent years.However,the objective functions for many optimization problems are non-convex,non-smooth,and even non-Lipschitz continuous.At present,there are few FISTAs to solve these problems.In this paper,we study a class of non-smooth convex optimization problems.The objective function is the sum of a smooth convex function and a non-smooth convex function.For such problems,an improved FISTA and restart FISTA are given.The main contents are summarized as follows:For a class of non-smooth convex optimization problems,combined with FISTA given by Beck and Teboulle,an improved FISTA is proposed.Improved FISTA selects the initial step 1/L_k to 1/L₀ at the beginning of the k th iteration.The step size chosen in this way can avoid the large Lipschitz constant of the improved FISTA in the initial stage of the iteration,and get a better iteration step.The improved FISTA is a non-monotonic algorithm and proves the convergence speed of the improved FISTA theoretically.The improved FISTA is applied to solve the Lasso problem,and the comparison is made from the aspects of running time,number of iterations and relative error.Numerical experiments show that improved FISTA is effective.Aiming at improving FISTA is a non-monotonic algorithm,combined with the restart technique given by Giselsson et al.,a restart FISTA for non-smooth convex optimization problem is proposed and the convergence speed of the algorithm is proved.At the same time,the restart conditions in the algorithm are analyzed.The restart FISTA and FISTA are applied to solve the Lasso problem.The numerical experiment results show that the restart FISTA is superior to FISTA in the convergence speed.The algorithm under different restart conditions is used to solve the Lasso problem.The numerical experiment results show that the restart FISTA with a restart condition of T2 is superior to the restart FISTA with a restart condition of T1 in the convergence speed.