我們要用迭代方法來研究最小化問題
min_{x∈C} Θ(x) (P)
, 其中C是在實Hilbert空間H上的非擴張映射T的固定點集合, 目標函數Θ:H→R連續Gateaux可微。 梯度投影法涉及投影算子P_{c}。 然而, 當C=Fix(T)時, 我們提供一種混合
(hybrid )迭代方法求解(P), 這種方法涉及到算子T, 而不再涉及到投影P_{c}。 我們用兩個例子來說明我們的混合迭代方法: 第一個例子, 目標函數Θ(x)=(1/2)||x-u||^2, 其中u∈C有一固定點; 第二個例子中, 目標函數
Θ(x)=<Ax,x> - <x,b>, 其中A為H上的一個強正有界線性算子, b為H中的一個固定點向量。

In this paper we study through iterative methods the minimization problem
min_{x&#8712;C} Θ(x) (P)
where the set C of constraints is the set of fixed points of a nonexpansive mapping T in a real Hilbert space H, and the objective function Θ:H→R is supposed to be continuously Gateaux dierentiable. The gradient projection method for solving problem (P) involves with the projection P_{C}. When C = Fix(T), we provide a
so-called hybrid iterative method for solving (P) and the method involves with the mapping T only. Two special cases are included: (1) Θ(x)=(1/2)||x-u||^2 and (2) Θ(x)=<Ax,x> - <x,b>. The first case corresponds to finding a fixed point of T which is closest to u from the fixed point set Fix(T). Both cases have received a lot of investigations recently.

1 Introduction 1
2 Preliminaries 3
3 The Algorithm and Its Convergence 5
4 Applications 17
References 19

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In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. New York: Elsevier, 2001, pp. 473504.