| Subgradient methods,proposed by Shor,popularized and developed by Albert,Iusem,Nesterov,Polyak,Solodov and other scholars,is now a common algorithm for solving nonsmooth(nondifferentiable)optimization problems.A main difference between smooth and nonsmooth optimization problems lies in the way of step size selection.For example,constant step-sizes no longer apply to nonsmooth objective functions.In the latter case,diminishing step-sizes must be used.In this paper,the projected subgradient algorithm in infinite-dimensional Hilbert space is studied.The main contributions of this paper are as follows:1.We provide a simple proof to the weak convergence theorem of Albert,at al for the projected subgradient algorithm.2.We apply a regularization technique(i.e.viscosity approximation method)to the projected subgradient algorithm to obtain two regularized projected subgradient algorithms,and moreover,we prove the strong convergence of these two algorithms.3.We provide the subgradient forcing strong convergence algorithm,which is an improvement of Bello-Cruz and Iusem’s algorithm,and prove its strong convergence.4.We develop the CQ algorithm for nonsmooth convex optimization problems to the projected subgradient CQ algorithm,and moreover,prove its strong convergence. |
