Vector Variational-Like Inequalities and Vector Optimization Problems Involving ρ-(η,d)-B Invexity on Riemannian Manifolds

LIU Shuang; MO Dingyong; ZHOU Zhiang

doi:10.21656/1000-0887.400227

Volume 41 Issue 4

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2020 > 41(4): 458-466

LIU Shuang, MO Dingyong, ZHOU Zhiang. Vector Variational-Like Inequalities and Vector Optimization Problems Involving ρ-(η,d)-B Invexity on Riemannian Manifolds[J]. Applied Mathematics and Mechanics, 2020, 41(4): 458-466. doi: 10.21656/1000-0887.400227

Citation:

PDF( 356 KB)

Vector Variational-Like Inequalities and Vector Optimization Problems Involving ρ-(η,d)-B Invexity on Riemannian Manifolds

doi: 10.21656/1000-0887.400227

¹College of Science, Chongqing University of Technology, Chongqing 400054, P.R.China;²Chongqing Yuzhong Insitute for Teacher Education, Chongqing 400015, P.R.China

Funds: The National Natural Science Foundation of China(11861002)

Received Date: 2019-07-27
Rev Recd Date: 2019-11-19
Publish Date: 2020-04-01

Abstract

Abstract

A class of optimality problems involving the generalized directional derivatives were studied on Riemannian manifolds. Firstly, by means of the generalized directional derivative, three concepts of the ρ-(η,d)-B invex function, the pseudo ρ-(η,d)-B invex function and the quasi ρ-(η,d)-B invex function on Riemannian manifolds were introduced. Secondly, the relations between the solution to variational inequalities and the solution to the optimization problem on Riemannian manifolds were discussed. Finally, the Kuhn-Tucker sufficient condition for the optimality problem was established.
- Riemannian manifold,
- vector variational inequality,
- vector optimization problem,
- ρ-(η,d)-B invexity

FullText(HTML)

References(20)

References

[1]	HANSON M A. On sufficiency of the Kuhn-Tucker conditions[J]. Journal of Mathematical Analysis and Applications,1981,80(2): 545-550.
[2]	GULATI T R, ISLAM M A. Sufficiency and duality in multiobjective programming involving generalized F -convex functions[J]. Journal of Mathematical Analysis and Applications,1994,183(1): 181-195.
[3]	CROUZEIX J P, LEGAZ J E M, VOLLE M. Generalized Convexity, Generalized Monotonicity: Recent Results [M]. Springer Science and Business Media, 2013.
[4]	CAMBINI A, MARTEIN L. Generalized Convexity and Optimization: Theory and Applications [M]. Springer Science and Business Media, 2008.
[5]	GUDDER S, SCHROECK F. Generalized convexity[J]. SIAM Journal on Mathematical Analysis,2012,11(6): 984-1001.
[6]	陈望, 周志昂. 基于改进集的带约束集值向量均衡问题的最优性条件[J]. 应用数学和力学, 2018,39(10): 1189-1197.(CHEN Wang, ZHOU Zhiang. Optimality conditions for set-valued vector equilibrium problems with constraints involving improvement sets[J]. Applied Mathematics and Mechanics,2018,39(10): 1189-1197.(in Chinese))
[7]	黄应全. 关于向量值 D -半预不变真拟凸映射的刻画[J]. 应用数学和力学, 2018,〖STHZ〗 39(3): 364-370.(HUANG Yingquan. Characterizations of D-properly semi-prequasi-invex mappings[J]. Applied Mathematics and Mechanics,2018,39(3): 364-370.(in Chinese))
[8]	杨玉红, 李飞. 非光滑半无限多目标优化问题的最优性充分条件[J]. 应用数学和力学, 2017,38(5): 526-538.(YANG Yuhong, LI Fei. Sufficient optimality conditions for nonsmooth semi-infinite multiobjective optimization problems[J]. Applied Mathematics and Mechanics,2017,38(5): 526-538.(in Chinese))
[9]	RAPCSAK T. Smooth Nonlinear Optimization in Rⁿ[M]. USA: Springer, 1997.
[10]	PINI R. Convexity along curves and indunvexity[J]. Optimization,1994,29(4): 301-309.
[11]	MITITELU S. Generalized invexity and vector optimization on differentiable manifolds[J]. Differential Geometry: Dynamical Systems,2001,3(1): 21-31.
[12]	BARANI A, POURYAYEVALI M R. Invex sets and preinvex functions on Riemannian manifolds[J]. Journal of Mathematical Analysis and Applications,2007,〖STHZ〗 328(2): 767-779.
[13]	AGARWAL R P, AHMAD I, IQBAL A, et al. Generalized invex sets and preinvex functions on Riemannian manifolds[J]. Taiwanese Journal of Mathematics,2012,16(5): 1719-1732.
[14]	CHEN S L, HUANG N J, O’REGAN D. Geodesic B -preinvex functions and multiobjective optimization problems on Riemannian manifolds[J]. Journal of Applied Mathematics,2014,2014: 1-12.
[15]	MISHRA S K, WANG S Y. Vector variational-like inequalities and non-smooth vector optimization problems[J]. Nonlinear Analysis: Theory, Methods & Applications,2006,64(9): 1939-1945.
[16]	肖刚, 肖红, 刘三阳. 黎曼流形上的向量似变分不等式与向量优化问题[J]. 安徽大学学报(自然科学版), 2009,33(3): 5-8.(XIAO Gang, XIAO Hong, LIU Sanyang. Vector variational-like inequalities and vector optimization problems on Riemannian manifolds[J]. Journal of Anhui University(Natural Science Edition),2009,33(3): 5-8.(in Chinese))
[17]	陈胜兰, 方长杰. 黎曼流形上的广义向量似变分不等式和向量优化问题[J]. 四川师范大学学报(自然科学版), 2016,39(3): 332-336.(CHEN Shenglan, FANG Changjie. Generalized vector variational-like inequality and vector optimization problem on Riemannian manifolds[J]. Journal of Sichuan Normal University(Natural Science),2016,39(3): 332-336.(in Chinese))
[18]	LI C, MORDUKHOVICH B S, WANG J H, et al. Weak sharp minima on Riemannian manifolds[J]. SIAM Journal on Optimization,2011,21(4): 1523-1560.
[19]	肖刚, 刘三阳. 黎曼流形上非可微多目标规划的必要最优性条件[J]. 吉林大学学报(理学版), 2008,〖STHZ〗 46(2): 209-213.(XIAO Gang, LIU Sanyang. Necessary optimality conditions of nondifferentiable-multiobjective programming on Riemannian manifolds[J]. Journal of Jilin University(Science Edition),2008,46(2): 209-213.(in Chinese))
[20]	AZAGRA D, FERRERA J, LOPEZMESAS F. Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds[J]. Journal of Functional Analysis,2003,220(2): 304-361.