Characterization of weakly sharp solutions of a variational-type inequality with convex functional

In this paper, we consider a variational-type inequality and study its weak sharp solutions in terms of a dual gap function, under certain assumptions and convex notion of a functional. Moreover, we aim to constitute the relationship between the minimum principle sufficiency property and weak sharp solutions of the considered variational-type inequality. A numerical result is also constructed to give a better insight for the main derived result.

[1]  M. Ferris,et al.  Weak sharp minima in mathematical programming , 1993 .

[2]  C. S. Lalitha,et al.  Vector variational inequalities with cone-pseudomonotone bifunctions , 2005 .

[3]  Suliman Al-Homidan,et al.  Minimum and maximum principle sufficiency properties for nonsmooth variational inequalities , 2016, Optim. Lett..

[4]  RELATIONS BETWEEN GENERALIZED VECTOR VARIATIONAL-LIKE INEQUALITIES AND VECTOR OPTIMIZATION PROBLEMS , 2012 .

[5]  J. Hiriart-Urruty,et al.  Fundamentals of Convex Analysis , 2004 .

[6]  M. A. Hanson,et al.  Duality for variational problems , 1967 .

[7]  Shengkun Zhu,et al.  Weak sharp efficiency in multiobjective optimization , 2016, Optim. Lett..

[8]  Marcin Studniarski,et al.  Weak sharp minima in multiobjective optimization , 2007 .

[9]  Suliman Al-Homidan,et al.  Finite convergence analysis and weak sharp solutions for variational inequalities , 2017, Optim. Lett..

[10]  M. A. Hanson.  On sufficiency of the Kuhn-Tucker conditions , 1981 .

[11]  Michael C. Ferris,et al.  Minimum principle sufficiency , 1992, Math. Program..

[12]  Soon-Yi Wu,et al.  Weak Sharp Solutions of Variational Inequalities in Hilbert Spaces , 2004, SIAM J. Optim..

[13]  X. Long,et al.  Gap functions and existence of solutions for generalized vector quasivariational inequalities , 2008 .

[14]  A. Mehra,et al.  Approximate convexity in vector optimisation , 2006, Bulletin of the Australian Mathematical Society.

[15]  Marius Durea,et al.  Existence conditions for generalized vector variational inequalities , 2011, Ann. Oper. Res..

[16]  Jafar Zafarani,et al.  Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces , 2013, Optim. Lett..

[17]  Shengjie Li,et al.  Duality and gap function for generalized multivalued ϵ-vector variational inequality , 2013 .

[18]  Moon Hee Kim,et al.  Relations between vector continuous-time program and vector variational-type inequality , 2004 .

[19]  Manuel Arana-Jiménez,et al.  On variational problems: Characterization of solutions and duality , 2005 .

[20]  Zili Wu,et al.  Characterization of weakly sharp solutions of a variational inequality by its primal gap function , 2016, Optim. Lett..

[21]  W. Song,et al.  Weak sharp solutions for variational inequalities in Banach spaces , 2011 .

[22]  Patrice Marcotte,et al.  Erratum: Weak Sharp Solutions of Variational Inequalities , 2000, SIAM J. Optim..

[23]  O. Mangasarian,et al.  NONLINEAR PERTURBATION OF LINEAR PROGRAMS , 1979 .

[24]  Lu-Chuan Ceng,et al.  Existence theorems for generalized vector variational inequalities with a variable ordering relation , 2010, J. Glob. Optim..

[25]  S. R. Mohan,et al.  On Invex Sets and Preinvex Functions , 1995 .

[26]  Xiaoqi Yang,et al.  Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities , 2002 .

[27]  M. A. Hanson,et al.  Bounds for functionally convex optimal control problems , 1964 .