Exponential type vector variational-like inequalities and nonsmooth vector optimization problems

详细信息    查看全文

• 作者：Anurag Jayswal ; Sarita Choudhury
• 关键词：Locally Lipschitz function ; Nonsmooth vector optimization problems ; Nonsmooth exponential type vector variational ; like inequalities ; (p ; r) ; invexity ; Fan ; KKM Theorem ; 90C26 ; 90C29 ; 49J52 ; 58E17 ; 58E35
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：49
• 期：1-2
• 页码：127-143
• 全文大小：425 KB
• 参考文献：1.Ahmad, M.K., Salahuddin, : Existence of solutions for generalized implicit vector variational-like inequalities. Nonlinear Anal. 67, 430鈥?41 (2007)MATH MathSciNet CrossRef
  2.Ansari, Q.H., Lee, G.M.: Nonsmooth vector optimization problems and Minty vector variational inequalities. J. Optim. Theory Appl. 145, 1鈥?6 (2010)MATH MathSciNet CrossRef
  3.Antczak, T.: \((p, r)\) -invexity in multiobjective programming. Eur. J. Oper. Res. 152, 72鈥?7 (2004)MATH MathSciNet CrossRef
  4.Antczak, T.: \(r\) -pre-invexity and \(r\) -invexity in mathematical programming. Comput. Math. Appl. 50, 551鈥?66 (2005)MATH MathSciNet CrossRef
  5.Clarke, F.H.: Optimization and nonsmooth analysis. Wiley-Interscience, New York (1983)MATH
  6.Fan, K.: A generalization of Tychonoffs fixed point theorem. Math. Ann. 142, 305鈥?10 (1961)MATH MathSciNet CrossRef
  7.Farajzadeh, A., Noor, M.A., Noor, K.I.: Vector nonsmooth variational-like inequalities and optimization problems. Nonlinear Anal. 71, 3471鈥?476 (2009)MATH MathSciNet CrossRef
  8.Giannessi, F. (ed.): Vector variational inequalities and vector equilibria. Math. Theories. Kluwer Academic Publishers, Dordrecht (2000)
  9.Ho, S.C., Lai, H.C.: Optimality and duality for nonsmooth minimax fractional programming problem with exponential \((p, r)\) -invexity. J. Nonlinear Convex Anal. 13, 433鈥?47 (2012)MATH MathSciNet
  10.Jayswal, A., Choudhury, S., Verma, R.U.: Exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities. J. Appl. Math. Comput. 45, 87鈥?7 (2014)MATH MathSciNet CrossRef
  11.Jeyakumar, V.: Equivalence of a saddle-point and optima, and duality for a class of non-smooth nonconvex problems. J. Math. Anal. Appl. 130, 334鈥?43 (1988)MATH MathSciNet CrossRef
  12.Liu, Z.B., Kim, J.K., Huang, N.J.: Existence of solutions for nonconvex and nonsmooth vector optimization problems. J. Inequal. Appl. 2008, Article ID 678014, p 7 (2008)
  13.Long, X.J., Quan, J., Wen, D.J.: Proper efficiency for set-valued optimization problems and vector variational-like inequalities. Bull. Korean Math. Soc. 50, 777鈥?86 (2013)MATH MathSciNet CrossRef
  14.Long, X.J., Peng, J.W., Wu, S.Y.: Generalized vector variational-like inequalities and nonsmooth vector optimization problems. Optimization 61, 1075鈥?086 (2012)MATH MathSciNet CrossRef
  15.Oveisiha, M., Zafarani, J.: Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces. Optim. Lett. 7, 709鈥?21 (2013)MATH MathSciNet CrossRef
  16.Reiland, T.W.: Nonsmooth invexity. Bull. Aust. Math. Soc. 42, 437鈥?46 (1990)MATH MathSciNet CrossRef
  17.Rezaie, M., Zafarani, J.: Vector optimization and variational-like inequalities. J. Glob. Optim. 43, 47鈥?6 (2009)MATH MathSciNet CrossRef
  18.Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton (1970)MATH
  19.Santos, L.B., Ruiz-Garz贸n, G., Rojas-Medar, M., Rufi谩n-Lizana, A.: Some relations between variational-like inequality problems and vectorial optimization problems in Banach spaces. Comput. Math. Appl. 55, 1808鈥?814 (2008)MATH MathSciNet CrossRef
  20.Zeng, L.C., Yao, J.C.: Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 36, 483鈥?97 (2006)MATH MathSciNet CrossRef
  
• 作者单位：Anurag Jayswal (1)
  Sarita Choudhury (1)
  
  1. Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826 004, Jharkhand, India
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Theory of Computation
  Mathematics of Computing
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1865-2085

文摘

This paper is devoted to study a new class of exponential form of vector variational-like inequality problems comprised of locally Lipschitz functions having exponential type invexities. In the setting of Banach space, we investigate the relationship of nonsmooth exponential type vector variational-like inequality problems with vector optimization problems involving nonsmooth \((p,r)\)-invex functions. Also, we explore the conditions for solvability of the aforesaid nonsmooth exponential type vector variational-like inequality problems using Fan-KKM Theorem. Moreover, we provide examples to elucidate our results. Keywords Locally Lipschitz function Nonsmooth vector optimization problems Nonsmooth exponential type vector variational-like inequalities (p, r)-invexity Fan-KKM Theorem