On nonsmooth V-invexity and vector variational-like inequalities in terms of the Michel–Penot subdifferentials

In this paper, we establish some results which exhibit an application for Michel–Penot subdifferential in nonsmooth vector optimization problems and vector variational-like inequalities. We formulate vector variational-like inequalities of Stampacchia and Minty type in terms of the Michel–Penot subdifferentials and use these variational-like inequalities as a tool to solve the vector optimization problem involving nonsmooth V-invex function. We also consider the corresponding weak versions of the vector variational-like inequalities and establish various results for the weak efficient solutions.

[1]  Q. H. Ansari,et al.  Generalized Minty Vector Variational-Like Inequalities and Vector Optimization Problems , 2009 .

[2]  Kok Lay Teo,et al.  Some Remarks on the Minty Vector Variational Inequality , 2004 .

[3]  Shashi Kant Mishra,et al.  Relationships between Optimization of V-invex Function and Vector Variational-like Inequalities , 2012 .

[4]  Kin Keung Lai,et al.  Generalized Convexity and Vector Optimization , 2008 .

[5]  Tamás Rapcsák,et al.  New Trends in Mathematical Programming , 1998 .

[6]  Shashi Kant Mishra,et al.  Invexity and Optimization , 2008 .

[7]  Xiaoqi Yang,et al.  Vector variational-like inequality with pseudoinvexity , 2006 .

[8]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[9]  S. Itoh,et al.  Variational inequalities and complementarity problems , 1978 .

[10]  Qamrul Hasan Ansari,et al.  Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities , 2010 .

[11]  Philippe Michel,et al.  A generalized derivative for calm and stable functions , 1992, Differential and Integral Equations.

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

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

[14]  N. D. Yen,et al.  Vector variational inequality as a tool for studying vector optimization problems , 1998 .

[15]  Shashi Kant Mishra,et al.  Nonsmooth minimax programming problems with V − r-invex functions , 2010 .

[16]  Shashi Kant Mishra,et al.  On V-r-invexity and vector variational-like inequalities , 2012 .

[17]  Tadeusz Antczak,et al.  Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity , 2009, J. Glob. Optim..

[18]  Jane J. Ye,et al.  Nondifferentiable Multiplier Rules for Optimization and Bilevel Optimization Problems , 2004, SIAM J. Optim..

[19]  R. Rockafellar.  Convex Analysis: (pms-28) , 1970 .

[20]  Jonathan M. Borwein,et al.  The differentiability of real functions on normed linear space using generalized subgradients , 1987 .

[21]  Douglass J. Wilde,et al.  Foundations of Optimization. , 1967 .

[22]  Vilfredo Pareto,et al.  Cours d'économie politique : professé à l'Université de Lausanne , 1896 .

[23]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[24]  Kin Keung Lai,et al.  Optimality and duality for nonsmooth multiobjective optimization problems with generalized V -r-invexity , 2010 .

[25]  Qamrul Hasan Ansari,et al.  Generalized vector variational-like inequalities and vector optimization , 2012, J. Glob. Optim..

[26]  Shashi Kant Mishra,et al.  On Generalized Minty and Stampacchia Vector Variational-Like Inequalities and V-Invex Vector Optimization in Asplund Spaces , 2013 .

[27]  Jane J. Ye,et al.  Multiplier Rules Under Mixed Assumptions of Differentiability and Lipschitz Continuity , 2000, SIAM J. Control. Optim..

[28]  A. Cambini,et al.  Generalized Convexity and Optimization: Theory and Applications , 2008 .

[29]  S. J. Li,et al.  On vector variational-like inequalities and set-valued optimization problems , 2011, Optim. Lett..

[30]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[31]  Kin Keung Lai,et al.  V-Invex Functions and Vector Optimization , 2007 .

[32]  Shashi Kant Mishra,et al.  On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities , 2013, J. Optim. Theory Appl..

[33]  Antonio Rufián-Lizana,et al.  Relationships between vector variational-like inequality and optimization problems , 2004, Eur. J. Oper. Res..

[34]  Jen-Chih Yao,et al.  Recent developments in vector optimization , 2012 .

[35]  Xiaoqi Yang,et al.  Generalized convex functions and vector variational inequalities , 1993 .

[36]  Jafar Zafarani,et al.  Generalized Invariant Monotonicity and Invexity of Non-differentiable Functions , 2006, J. Glob. Optim..

[37]  Kok Lay Teo,et al.  Generalized Invexity and Generalized Invariant Monotonicity , 2003 .

[38]  Vaithilingam Jeyakumar,et al.  On generalised convex mathematical programming , 1992, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[39]  Kok Lay Teo,et al.  Characterizations and Applications of Prequasi-Invex Functions , 2001 .

[40]  Shashi Kant Mishra,et al.  Vector variational-like inequalities and non-smooth vector optimization problems , 2006 .

[41]  F. Giannessi.  On Minty Variational Principle , 1998 .

[42]  R. Weiner.  Lecture Notes in Economics and Mathematical Systems , 1985 .

[43]  Shashi Kant Mishra,et al.  A Note on the Paper “On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities” , 2013, J. Optim. Theory Appl..

[44]  B. D. Craven,et al.  Nondifferentiable optimization by smooth approximations , 1986 .