A Class Of Gradient Algorithms With L₁ Norm Optimization Problems

The optimization problem with l₁-norm is a kind of important nonsmooth optimization problem.It is widely used in many fileds,such as compressed sensing filed and signal reconstruction filed.In recent years,the theory and methods for the optimization problem with l₁ -norm have been developed greatly.In this paper,we study the methods for solving the optimization problem with l₁ -norm.We give the smoothing FR conjugate gradient method and the smoothing three-term conjugate gradient method.The structure and main contents of this paper are summarized as follows:In the first chapter,we introduce the form and basic situation of the optimization problem with l₁ -norm.We also give the significance of the optimization problem with l₁-norm.In the second chapter,the smoothing FR conjugate gradient method for solving the optimization problem with l₁-norm is given.Firstly,we give the basic form of the optimization problem withl₁-norm.The optimization problem withl₁-norm is transformed into a unconstrained optimization problem.Then,the smooth approximation function is used to solve the unconstrained optimization problem.The optimization problem with l₁ -norm is transformed to a smooth unconstrained optimization problem.And under the general assumptions,the global convergence of the smoothing FR conjugate gradient method are also given.Finally,the numerical results indicate that the given method is effective.In the third chapter,the smoothing three-term conjugate gradient method for solving the optimization problem with l₁ -norm is given.The smoothing three-term conjugate gradient method is proposed to solve the transformed smooth unconstrained optimization problem.Under general assumptions,the global convergence of the smoothing three-term conjugate gradient method is given.Finally,the related numerical results indicate the efficiency of the method.