A Regularized Gradient Projection Method for the Minimization Problem

Abstract

We investigate the following regularized gradient projection algorithmxn+1=Pc(I−γn(∇f+αnI))xn,n≥0. Under some different control conditions, we prove that this gradient projection algorithm strongly converges to the minimum norm solution of the minimization problemminx∈Cf(x).

Approximation of Solutions to Constrained Convex Minimization Problem in Hilbert Spaces

“…Also, our result does not require additional projections which was required in Theorem 5.4 of Xu [22] and Theorem 3.3 of [24] in order to guarantee strong convergence. Our method of proof is different from the methods of proof of Xu [22], Yao et al [23], Ceng et al [6], Su and Xu [18], and others.…”

“…The gradient projection (or projected-gradient) algorithm is a powerful tool for solving constrained convex optimization problems and has been extensively studied (see [5,6,9,11,15,[17][18][19][20][21][22][23] and the references therein). It has been recently applied to solve split feasibility problems which find applications in image reconstructions and the intensity-modulated radiation therapy (see [3,4,14,25]).…”

On The Strong Convergence of The Gradient Projection Algorithm with Tikhonov regularizing term

“…In the general case, the convergence of the sequence (x n ) for the discreet algorithm (GP) and the trajectory x(t) for the continuous dynamical system (CGP) are only weak (see [4] and [3]) and the corresponding limits are an undefined minimizers of f over Q which may depend on the initial data x 0 . To overcome these two weakness many modifications of the algorithm (GP) and its continuous version (CGP) are proposed [4,[7][8][9][10][11][12]. For instance, in 2002, J. Bolte [8] considered the following dynamical system…”

Solving a Split Feasibility Problem by the Strong Convergence of Two Projection Algorithms in Hilbert Spaces

“…In this scene, we do not mean the sum of A and B in the iterates, but each step of iterates includes only A as the forward term and B as the backward term. As special cases, this technique gets involved heavily in a study of the proximal point algorithm [11][12][13] and the gradient method [14][15][16][17].…”