关于伪单调变分不等式与不动点问题的新投影算法

摘要/Abstract

摘要：

该文在Hilbert空间中给出了一个新投影算法，找到了伪单调变分不等式问题的解集与半压缩映射的不动点集的公共元.在映射是伪单调和一致连续的条件下，证明了强收敛定理.数值实验结果表明了新算法的有效性和优越性.

Abstract:

In this paper, we propose a new projection algorithm for finding a common element of psedomonotone variational inequality problems and fixed point set of demicontractive mappings in Hilbert spaces. We prove that this new algorithm converges strongly to the common element for a psedomonotone and uniformly continuous mapping. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the new projection algorithm.

Key words: Variational inequality, Fixed point, Projection algorithm, Uniformly continuous, Pseudomonotone

中图分类号:

O224

杨静,龙宪军. 关于伪单调变分不等式与不动点问题的新投影算法[J]. 数学物理学报, 2022, 42(3): 904-919.

Jing Yang,Xianjun Long. A New Projection Algorithm for Solving Pseudo-Monotone Variational Inequality and Fixed Point Problems[J]. Acta mathematica scientia,Series A, 2022, 42(3): 904-919.

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参考文献 21

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$f(x)$	$m=30$		$m=60$		$m=90$
$f(x)$	Iter	CPU time	Iter	CPU time	Iter	GPU time
$\frac{x}{4}$	51	0.8511	52	0.7227	52	0.7869
$\frac{x}{2}$	53	0.7082	58	0.8453	62	0.9145
$\frac{\sin(x)}{2}$	55	0.7157	53	0.6831	57	0.9798
$\frac{\cos(x)}{2}$	236	5.7441	318	9.4689	427	19.516

$err=10^{-8}$	$m=50$		$m=60$		$m=70$
$err=10^{-8}$	Iter	CPU time	Iter	CPU time	Iter	CPU time
Alg 3.1	41	0.3043	42	0.3151	42	0.9606
Alg 1 in [8]	136	0.9405	145	0.9811	1902	38.337
Alg 2 in [8]	329	2.5770	490	3.7967	527	4.3241
Alg 4 in [7]	237	3.2041	533	9.0855	6513	106.62

$err$	Alg.3.1		Alg.3.2 in [12]		Alg.3.1 in [12]
$err$	Iter	CPU time	Iter	CPU time	Iter	GPU time
$10^{-10}$	12	0.0144	17	0.0239	24868	0.2387
$10^{-12}$	14	0.0154	19	0.0243	$\geq 10^{6}$	1.7069
$10^{-14}$	16	0.0165	22	0.0297	$\geq 10^{7}$	15.944

	$10^{-5}$		$10^{-10}$		$10^{-15}$
	Iter	CPU time	Iter	CPU time	Iter	GPU time
Alg 3.1	28	0.0138	66	0.0163	104	0.0529
Alg 1 in [8]	39	0.0238	80	0.0274	120	0.0635
Alg 2 in [8]	45	0.0257	126	0.0357	207	0.0933
Alg 3.1 in [13]	42	0.0142	86	0.0175	127	0.0548

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