考慮有約束條件情況下凸的最小化問題:
min_x∈C f(x)
在本篇論文中我們提供梯度投影法來產生序列x^k，根據下列的迭代方法
x^(k+1) = P_c(x^k − α_k∇f(x^k)), k= 0, 1, · · · ,
我們基本的想法是將最小化問題轉換成一個固定點的演算法:
x^(k+1) = T_(αk)x^k, k = 0, 1, · · ·
以此來解決最小化問題.
本篇文章中我們提供了梯度投影法根據不同步長的選擇去討論其解的收斂問題.

We consider the constrained convex minimization problem:
min_x∈C f(x)
we will present gradient projection method which generates a sequence x^k
according to the formula
x^(k+1) = P_c(x^k − α_k∇f(x^k)), k= 0, 1, · · · ,
our ideal is rewritten the formula as a xed point algorithm:
x^(k+1) = T_(αk)x^k, k = 0, 1, · · ·



is used to solve the minimization problem.
In this paper, we present the gradient projection method(GPM) and different choices of the stepsize to discuss the convergence of gradient projection
method which converge to a solution of the concerned problem.

Contents
`Š i
Abstract ii
1 Introduction 1
2 Preliminaries 4
2.1 Nonexpansive mappings . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Monotone operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 The Gradient-Projection Algorithm And It's Convergence 7
3.1 Variable Stepsize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Strongly Monotone Gradient . . . . . . . . . . . . . . . . . . . . . . 10
Reference 15

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