一般约束极大极小优化问题一个强收敛的广义梯度投影算法

Abstract

Abstract:

In this paper, minimax optimization problems with inequality and equality constraints is discussed. The original problem is transformed into an associated simple problem with a penalty term and only inequality constraints, then a new generalized gradient projection algorithm is presented. The main characters of the proposed algorithm are as follows:the improved search direction is generated by only one generalized gradient projection explicit formula; the new optimal identification function is introduced; the algorithm is globally and strongly convergent under some mild assumptions. Finally, the numerical results show that the proposed algorithm is promising.

Key words: Nonlinear general constraints, Minimax optimization problems, Generalized gradient projection method, Global convergence, Strong convergence

CLC Number:

O221.2

Guodong Ma. A Strongly Convergent Generalized Gradient Projection Method for Minimax Optimization with General Constraints[J].Acta mathematica scientia,Series A, 2020, 40(3): 641-649.

Trendmd

Figures/Tables 1

References 16

1	Yu Y H , Gao L . Nonmonotone line search algorithm for constrained minimax problems. J Optimiz Theory App, 2002, 115 (2): 419- 446 doi: 10.1023/A:1020896407415
2	Jian J B , Zhang X L , Quan R , Ma Q . Generalized monotone line search SQP algorithm for constrained minimax problems. Optimization, 2009, 58 (1): 101- 131
3	Jian J B , Hu Q J , Tang C M . Superlinearly convergent norm-relaxed SQP method based on active set identification and new line search for constrained minimax problems. J Optimiz Theory App, 2014, 163 (3): 859- 883 doi: 10.1007/s10957-013-0503-5
4	Wang F S , Wang C L . An adaptive nonmonotone trust region method with curvilinear search for minimax problem. Appl Math Comput, 2013, 219, 8033- 8041
5	Ye F , Liu H W , Zhou S S , et al. A smoothing trust region Newton-CG method for minimax problem. Appl Math Comput, 2008, 199, 581- 589
6	Chi X N , Wei H J , Wan Z P , et al. Smoothing Newton algorithm for the circular cone programming with a nonmonotone line search. Acta Math Sci, 2017, 37B (5): 1262- 1280
7	Obasanjo E , Tzallas-Regas G , Rustem B . An interior point algorithm for nonlinear minimax problems. J Optimiz Theory App, 2010, 144 (2): 291- 318 doi: 10.1007/s10957-009-9599-z
8	Mayne D Q , Polak E . Feasible directions algorithms for optimization problems with equality and inequality constraints. Math Program, 1976, 11 (1): 67- 80 doi: 10.1007/BF01580371
9	Jian J B , Xu Q J , Han D L . A strongly convergent norm-relaxed method of strongly sub-feasible direction for optimization with nonlinear equality and inequality constraints. Appl Math Comput, 2006, 182, 854- 870
10	Jian J B , Guo C H , Yang L F . A new generalized projection method of strongly sub-feasible directions for general constrained optimization. Pac J Optim, 2009, 5 (3): 507- 523
11	Rosen J B . The gradient projection method for nonlinear programming. Part I. Linear constraints. J Soc Ind Appl Math, 1960, 8 (1): 181- 217
12	Rosen J B . The gradient projection method for nonlinear programming. Part Ⅱ. Linear constraints. J Soc Ind Appl Math, 1961, 9 (4): 514- 532 doi: 10.1137/0109044
13	朱志斌, 王硕, 简金宝. 非线性优化一个超线性收敛的广义投影型可行方向法. 应用数学学报, 2014, 37 (1): 179- 192
	Zhu Z B , Wang S , Jian J B . A generalized projection feasible method with superlinear convergence for nonlinear optimization. Acta Math Appl Sin, 2014, 37 (1): 179- 192
14	简金宝. 光滑约束优化快速算法-理论分析与数值试验. 北京: 科学出版社, 2010
	Jian J B . Fast Algorithms for Smooth Constrained Optimization-Theoretical Analysis and Numerical Experiments. Beijing: Science Press, 2010
15	Ma G D , Jian J B . An ε-generalized gradient projection method for nonlinear minimax problems. Nonlinear Dynam, 2014, 75 (4): 693- 700 doi: 10.1007/s11071-013-1095-1
16	Ma G D , Zhang Y F , Liu M X . A generalized gradient projection method based on a new working set for minimax optimization problems with inequality constraints. J Inequal Appl, 2017, 2017, 51 doi: 10.1186/s13660-017-1321-3

问题	$n/l$/$m'$$/m$	$x^0$	算法	Ni	$F(x^*)$
问题1	$2/3/0/1$	$(-1, 3)^T$	算法A	$14$	$2.1266e-10$
			算法JZQM	$12$	$2.0079e-10$
			算法YG	$13$	$0$
问题2	$2/3/0/1$	$(-1, 1)^T$	算法A	$8$	$-1$
			算法JZQM	$15$	$-1$
			算法YG	$11$	$-1$
问题3	$2/6/2/0$	$(1, 3)^T$	算法A	$20$	$0.6256$
			算法JZQM	$15$	$0.5832$
			算法YG	$18$	$0.6164$
问题5	$2/2/0/1$	$(-1, 4)^T$	算法A	$20$	$-5.8714$
			算法JZQM	$12$	$-5.8754$
问题6	$3/3/1/2$	$(2, 3, 2)^T$	算法A	$8$	$-3.8621$
			算法JZQM	$5$	$-3.9345$
问题7	$10/8/0/3$	$(0, 1, 1, 1, 0, 0, 1, 1, 1, 0)^T$	算法A	$74$	$2.3521e+003$
			算法JZQM	$52$	$2.3339e+003$

[1]	Gang Cai. Viscosity Implicit Algorithms for a Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2020, 40(2): 395-407.
[2]	Cai Gang, Yekini Shehu. Viscosity Iterative Algorithm for Variational Inequality Problems and Fixed Point Problems of Strict Pseudo-Contractions in q-Uniformly Smooth and Uniformly Convex Banach Spaces [J]. Acta mathematica scientia,Series A, 2016, 36(4): 623-638.
[3]	Liu Yuanxing, Zhou Haiyun, Wang Peiyuan, Zhou Yu. Three Kinds of Iterative Algorithms for Solving the Constrained Split Common Fixed Point Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1115-1126.
[4]	Wang Yuehu, Zhang Congjun. Strong Convergence Based on Generalized Wiener-Hopf Equation Techniques for Solving Generalized Equilibrium Problems [J]. Acta mathematica scientia,Series A, 2015, 35(4): 695-709.
[5]	Cai Gang. Strong Convergence Theorems by A Viscosity Iterative Method for Fixed Point Problems of Nonexpansive Mappings in Banach Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(3): 487-502.
[6]	WAN Cheng-Gao. The Limit Theorems for Function of Markov Chains in Random Environments [J]. Acta mathematica scientia,Series A, 2015, 35(1): 163-171.
[7]	TAO Bao. Asymptotic Properties of the Negative Extreme-value Index Estimator [J]. Acta mathematica scientia,Series A, 2014, 34(3): 611-618.
[8]	CAI Gang, BU Shang-Quan. Strong Convergence Theorems of a Modified Mann Iterative Method for Nonexpansive Mappings in Banach Spaces [J]. Acta mathematica scientia,Series A, 2014, 34(1): 1-8.
[9]	CHEN Jia-Wei, WAN Zhong-Peng, ZHAO Lie-Ji. Convergence Analysis for Countable Family of Weak Bregman Relatively Nonexpansive Mappings [J]. Acta mathematica scientia,Series A, 2014, 34(1): 70-79.
[10]	HU Hui-Ying, ZENG Liu-Chuan. The New Iterative Scheme for two Equilibrium Problems and Fixed Point Problems of Infinitely Nonexpansive Mappings [J]. Acta mathematica scientia,Series A, 2013, 33(5): 860-881.
[11]	HU Chao-Ming, WAN Zhong, WANG Xu. A New Nonmonotone Spectral Conjugate Gradient Algorithm [J]. Acta mathematica scientia,Series A, 2013, 33(1): 78-88.
[12]	TANG Jing-Yong, HE Guo-Ping. A One-step Smoothing Newton Method for Second-order Cone Programming [J]. Acta mathematica scientia,Series A, 2012, 32(4): 768-778.
[13]	YANG Li, LI Jun. The CQ Method for Common \|Fixed Point Involving Asymptotically Nonexpansive Semigroups [J]. Acta mathematica scientia,Series A, 2010, 30(4): 1158-1165.
[14]	YIN Chang-Ming, LI Yong-Ming, WANG Peng-Yan. Strong Convergence Rates of the Maximum Quasi-Likelihood Estimator in Generalized Linear Models [J]. Acta mathematica scientia,Series A, 2009, 29(4): 1058-1064.
[15]	SHU Lan-Ping, LI Gang. Fixed Point \|Theorems for Asymptotically Nonexpansive Type Semigroups in General Banach Spaces [J]. Acta mathematica scientia,Series A, 2009, 29(2): 290-296.

A Strongly Convergent Generalized Gradient Projection Method for Minimax Optimization with General Constraints

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 1

References 16

Related Articles 15

Metrics

Comments

Recommended 10