Gap functions and error bounds for weak vector variational inequalities

References

• Aubin , JP and Ekeland , I . 1984 . Applied Nonlinear Analysis , New York : John Wiley and Sons .
   Google Scholar
• Aussel , D , Correa , R and Marechal , M . 2011 . Gap functions for quasivariational inequalities and generalized Nash equilibrium problems . J. Optim. Theory Appl. , 151 : 474 – 488 .
   Web of Science ®Google Scholar
• Aussel , D and Dutta , J . 2011 . On gap functions for multivalued Stampacchis variational inequalities . J. Optim. Theory Appl. , 149 : 513 – 527 .
   Web of Science ®Google Scholar
• Bianchi , M and Pini , R . 2006 . Sensitivity for parametric vector equilibria . Optimization , 55 : 221 – 230 .
   Web of Science ®Google Scholar
• Charitha , C and Dutta , J . 2010 . Regularized gap functions and error bounds for vector variational inequalities . Pac. J. Optim. , 6 : 497 – 510 .
   Web of Science ®Google Scholar
• Chen , CR , Li , SJ and Fang , ZM . 2010 . On the solution semicontinuity to a parametric generalized vector quasivariational inequality . Comput. Math. Appl. , 60 : 2417 – 2425 .
   Web of Science ®Google Scholar
• Chen , GY . 1992 . Existence of solutions for a vector variational inequality: An extension of Hartmann-Stampacchia theorem . J. Optim. Theory Appl. , 74 : 445 – 456 .
   Web of Science ®Google Scholar
• Chen , GY , Goh , CJ and Yang , XQ . 2000 . “ On gap functions for vector variational inequalities ” . In Vector Variational Inequalities and Vector Equilibria, Mathematical Theories , Edited by: Giannessi , F . 55 – 72 . Dordrecht : Kluwer Academic .
   Google Scholar
• Chen , GY , Huang , XX and Yang , XQ . 2005 . Vector Optimization: Set-valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems , Vol. 541 , Berlin : Springer-Verlag .
   Google Scholar
• Facchinei , F and Pang , JS . 2003 . Finite-dimensional Variational Inequalities and Complimentarity Problems , Vol. 1 , Berlin : Springer .
   Google Scholar
• Giannessi , F . 1980 . “ Theorems of the alternative, quadratic programs, and complementarity problems ” . In Variational Inequality and Complementarity Problems , Edited by: Cottle , RW , Giannessi , F and Lions , JL . 151 – 186 . Chichester : Wiley .
   Google Scholar
• Giannessi , F . 2000 . Vector Variational Inequalites and Vector Equilibrium , Dordrecht : Kluwer Academic Pubishers .
   Google Scholar
• Giannessi , F . 2005 . Constrained Optimization and Image Space Analysis, Volume 1 of Separation of Sets and Optimality Conditions , New York : Springer .
   Google Scholar
• Giannessi , F and Mastroeni , G . 2008 . Separation of sets and Wolfe duality . J. Glob. Optim. , 42 : 401 – 412 .
   Web of Science ®Google Scholar
• Gupta , R and Mehra , A . 2012 . Gap functions and error bounds for quasi variational inequalities . J. Glob. Optim. , 53 : 737 – 748 .
   Web of Science ®Google Scholar
• Hiriart-Urruty , JB . 1979 . Tangent cone, generalized gradients and mathematical programming in Banach spaces . Math. Oper. Res. , 4 : 79 – 97 .
   Google Scholar
• Huang , LR and Ng , KF . 2005 . Equivalent optimization formulations and error bounds for variational inequality problems . J. Optim. Theory Appl. , 125 : 299 – 314 .
   Web of Science ®Google Scholar
• Konnov , IV . 2005 . A scalarization approach for vector variational inequalities with applications . J. Glob. Optim. , 32 : 517 – 527 .
   Web of Science ®Google Scholar
• Li , G and Ng , KF . 2009 . Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems . SIAM J. Optim. , 20 : 667 – 690 .
   Web of Science ®Google Scholar
• Li , J and Huang , NJ . 2007 . An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems . J. Global Optim. , 39 : 247 – 260 .
   Web of Science ®Google Scholar
• Li , J and Huang , NJ . 2012 . Image space analysis for variational inequalities with cone constraints applications to traffic equilibria . Sci. China Math. , 55 : 851 – 868 .
   Web of Science ®Google Scholar
• Li , J , Huang , NJ and Yang , XQ . 2010 . Weak sharp minima for set-valued vector variational inequalities with an application . Eur. J. Oper. Res. , 205 : 262 – 272 .
   Web of Science ®Google Scholar
• Li , J and Mastroeni , G . 2010 . Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions . J. Optim. Theory Appl. , 145 : 355 – 372 .
   Web of Science ®Google Scholar
• Li , SJ , Teo , KL , Yang , XQ and Wu , SY . 2006 . Gap functions and existence of solutions to generalized vector quasi-equilibrium problems . J. Global Optim. , 34 : 427 – 440 .
   Web of Science ®Google Scholar
• Li , SJ , Xu , YD and Zhu , SK . 2012 . Nonlinear separation approach to constrained extremum problems . J. Optim. Theory Appl. , 154 : 842 – 856 .
   Web of Science ®Google Scholar
• Li , SJ , Yan , H and Chen , GY . 2003 . Differential and sensitivity properties of gap functions for vector variational inequalities . Math. Meth. Oper. Res. , 57 : 377 – 391 .
   Web of Science ®Google Scholar
• Mastroeni , G . 2003 . Gap functions for equilibrium problems . J. Global Optim. , 27 : 411 – 426 .
   Web of Science ®Google Scholar
• Mastroeni , G . 2010 . Some applications of the image space analysis to the duality theory for constrained extremum problems . J. Global Optim. , 46 : 603 – 614 .
   Web of Science ®Google Scholar
• Mastroeni , G . 2011 . Nonlinear separation in the image space with applications to penalty methods . Appl. Anal. , doi: 10.1080/00036811.2011.614603
   Web of Science ®Google Scholar
• Mastroeni , G , Panicucci , B , Passacantando , M and Yao , JC . 2009 . A separation approach to vector quasi-equilibrium problems: Saddle point and gap function . Taiwan. J. Math. , 13 : 657 – 673 .
   Web of Science ®Google Scholar
• Mastroeni , G and Pellegrini , L . 2011 . Conic separation for vector optimization problems . Optimization , 60 : 129 – 142 .
   Web of Science ®Google Scholar
• Meng , KW and Li , SJ . 2008 . Differential and sensitivity properties of gap functions for Minty vector variational inequalities . J. Math. Anal. Appl. , 337 : 386 – 398 .
   Web of Science ®Google Scholar
• Tardella , F . 1989 . On the image of a constrained extremum problem and some applications to existence of a minimum . J. Optim. Theory Appl. , 69 : 93 – 104 .
   Web of Science ®Google Scholar
• Yang , XQ and Yao , JC . 2003 . On the gap functions of prevariational inequalities . J. Optim. Theory Appl. , 116 : 437 – 452 .
   Web of Science ®Google Scholar
• Zaffaroni , A . 2003 . Degrees of efficiency and degrees of minimality . SIAM J. Control Optim. , 42 : 1071 – 1086 .
   Web of Science ®Google Scholar