A Regularized Gradient Projection Method for the Minimization Problem

Yonghong Yao,Wu Jigang,Shin Min Kang,Pei-Xia Yang

doi:10.1155/2012/259813

Yonghong Yao, Wu Jigang + Show 2 more

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https://doi.org/10.1155/2012/259813

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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).

Highlights

Let C be a nonempty closed and convex subset of a real Hilbert space H
Note that 1.2 can be rewritten as x∗ − x∗ − ∇f x∗, x − x∗ ≥ 0, ∀x ∈ C. This shows that the minimization 1.1 is equivalent to the fixed point problem
The gradient-projection algorithm 1.5 is a powerful tool for solving constrained convex optimization problems and has well been studied in the case of constant stepsizes γn γ for all n

Summary

Introduction

Let C be a nonempty closed and convex subset of a real Hilbert space H. PC x∗ − γ ∇f x∗ x∗, 1.4 where γ > 0 is any constant and PC is the nearest point projection from H onto C By using this relationship, the gradient-projection algorithm is usually applied to solve the minimization problem 1.1. The gradient-projection algorithm 1.5 is a powerful tool for solving constrained convex optimization problems and has well been studied in the case of constant stepsizes γn γ for all n. The reader can refer to 1–9 and the references therein It is known 3 that if f has a Lipschitz continuous and strongly monotone gradient, the sequence {xn} can be strongly convergent to a minimizer of f in C. Under some different control conditions, we prove that this gradient projection algorithm strongly converges to the minimum norm solution of the minimization problem 1.1

Preliminaries

Main Result