Nonlinear separation methods and applications for vector equilibrium problems using improvement sets

References

• Luc DT. Theory of vector optimization. Lecture notes in economics and mathematical systems. Berlin: Springer; 1989.
   Google Scholar
• Jahn J. Vector optimization theory, application and extensions. 2nd ed. Berlin: Springer; 2011.
   Google Scholar
• Chen GY, Huang XX, Yang XQ. Vector optimization: set-valued and variational analysis. Berlin: Springer; 2005.
   Google Scholar
• Borwein JM. Proper efficient points for maximizations with respect to cones. SIAM J Control Optim. 1997;15:57–63.
   Google Scholar
• Benson B. An improved definition of proper efficiency for vector maximizations with respect to cones. J Math Anal Appl. 1997;71:232–241.
   Google Scholar
• Henig MI. Proper efficiency with respect to cones. J Optim Theory Appl. 1982;36:387–407.
   Web of Science ®Google Scholar
• Rong WD, Wu YN. ε-Weak minimal solutions of vector optimization problems with set-valued maps. J Optim Theory Appl. 2000;106:569–579.
   Web of Science ®Google Scholar
• Yang XM. Near-subconvexlikeness in vector optimization with set-valued functions. J Optim Theory Appl. 2001;110:413–427.
   Web of Science ®Google Scholar
• Li JL. Constrained ordered equilibrium problems and applications. J Nonlinear Var Anal. 2017;1:357–365.
   Web of Science ®Google Scholar
• Mai TT, Luu DV. Optimality conditions for weakly efficient solutions of vector variational inequalities via convexificators. J Nonlinear Var Anal. 2018;2:379–389.
   Web of Science ®Google Scholar
• Chicco M, Mignanego F, Pusillo L, et al. Vector optimization problems via improvement sets. J Optim Theory Appl. 2011;150:516–529.
   Web of Science ®Google Scholar
• Gutiérrez C, Jiménez B, Novo V. Improvement sets and vector optimization. Eur J Oper Res. 2012;203:304–311.
   Google Scholar
• Zhao KQ, Yang XM. A unified stability result with perturbations in vector optimization. Optim Lett. 2013;7:1913–1919.
   Web of Science ®Google Scholar
• Zhao KQ, Yang XM. E-Benson proper efficiency in vector optimization. Optimization. 2015;64:739–752.
   Web of Science ®Google Scholar
• Zhao KQ, Chen GY, Yang XM. Approximate proper efficiency in vector optimization. Optimization. 2015;64:1777–1793.
   Web of Science ®Google Scholar
• Oppezzi P, Rossi A. Improvement sets and convergence of optimal points. J Optim Theory Appl. 2015;165:405–419.
   Web of Science ®Google Scholar
• Lalitha CS, Chatterjee P. Stability and scalarization in vector optimization using improvement sets. J Optim Theory Appl. 2015;166:825–843.
   Web of Science ®Google Scholar
• You MX, Li SJ. Nonlinear separation concerning E-optimal solution of constrained multi-objective optimization problems. Optim Lett. 2018;12:123–136.
   Web of Science ®Google Scholar
• Chen JW, Huang L, Li SJ. Separations and optimality of constrained multiobjective optimization via improvement sets. J Optim Theory Appl. 2018;178:794–823.
   Web of Science ®Google Scholar
• Zhou ZA, Wang C, Yang XM. Scalarizations and optimality of constraints setvalued optimization using improvement sets and image space analysis. J Optim Theory Appl. 2019. doi:10.1007/s10957-019-01554-3.
   Web of Science ®Google Scholar
• Giannessi F. Constrained optimization and image space analysis. Vol. 1: separation of sets and optimality conditions. New York: Springer; 2005.
   Google Scholar
• Chinaie M, Zafarani J. Image space analysis and scalarization of multivalued optimization. J Optim Theory Appl. 2009;142:451–467.
   Web of Science ®Google Scholar
• Li J, Huang NJ. Image space analysis for vector variational inequalities with matrix inequality constraints and applications. J Optim Theory Appl. 2010;145:459–477.
   Web of Science ®Google Scholar
• Luo HZ, Wu HX, JZ Liu. On saddle points in semidefinite optimization via separation scheme. J Optim Theory Appl. 2015;165:113–150.
   Web of Science ®Google Scholar
• Chen JW, Li SJ, Wan ZP, et al. Vector variational-like inequalities with constraints: separation and alternative. J Optim Theory Appl. 2015;166:460–479.
   Web of Science ®Google Scholar
• Chen JW, Köbis E, Köbis M, et al. Image space analysis for constrained inverse vector variational inequalities via multiobjective optimization. J Optim Theory Appl. 2018;177:816–834.
   Web of Science ®Google Scholar
• You MX, Li SJ. Characterization of duality for a generalized quasi-equilibrium problem. Appl Anal. 2018;97:1611–1627.
   Web of Science ®Google Scholar
• Li GH, Li SJ. Optimality conditions for vector optimization problems with non-cone constraints in image space. Appl Anal. 2018. doi:10.1080/00036811.2018.1506105.
   Web of Science ®Google Scholar
• Pellegrini L, Zhu SK. Constrained extremum problems, regularity conditions and image space analysis. Part II: the vector finite-dimensional case. J Optim Theory Appl. 2018;177:788–810.
   Web of Science ®Google Scholar
• Antoni C, Giannessi F. Images and existence of constrained scalar and vector extrema. J Optim Theory Appl. 2018;177:865–888.
   Web of Science ®Google Scholar
• Chen CR, Zuo X, Lu F, et al. Vector equilibrium problems under improvement sets and linear scalarization with stability applications. Optim Methods Soft. 2016;31:1240–1257.
   Web of Science ®Google Scholar
• Ziad A. Pure-strategy ε-Nash equilibrium in n-person nonzero-sum discontinuous games. Games Econom Behav. 1997;20:238–249.
   Web of Science ®Google Scholar
• Patrone F, Pusillo L, Tijs S. Multicriteria games and potentials. Top. 2007;15:138–145.
   Web of Science ®Google Scholar
• Li J, Mastroeni G. Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions. J Optim Theory Appl. 2010;145:355–372.
   Web of Science ®Google Scholar
• Xu YD, Li SJ. Gap functions and error bounds for weak vector variational inequalities. Optimization. 2014;63:1339–1352.
   Web of Science ®Google Scholar
• Dutta J, Kesarwani J, Gupta S. Gap functions and error bounds for nonsmooth convex vector optimization problem. Optimization. 2017;66:1807–1836.
   Web of Science ®Google Scholar
• Xu YD, Zhang PP. Gap functions for constrained vector variational inequalities with applications. Optimization. 2017;66:2171–2191.
   Web of Science ®Google Scholar
• Mastroeni G, Panicucci B, Passacantando M, et al. A separation approach to vector quasi-equilibrium problems: saddle point and gap function. Taiwan J Math. 2009;13:657–673.
   Web of Science ®Google Scholar
• Khan SA, Chen JW. Gap functions and error bounds for generalized mixed vector equilibrium problems. J Optim Theory Appl. 2015;166:767–776.
   Web of Science ®Google Scholar
• Li GH, Li SJ. Saddle points and gap functions for weak generalized Ky Fan inequalities. Optim Lett. 2018;12:1265–1280.
   Web of Science ®Google Scholar
• Hiriart-Urruty JB. Tangent cone, generalized gradients and mathematical programming in Banach spaces. Math Oper Res. 1979;4:79–97.
   Google Scholar
• Shapiro A, Scheinberg K. Duality and optimality conditions. In: Wolkowicz H, Saigal R, Vandenberghe L, editors. Handbook of semidefinite programming: theory, algorithms and applications, Vol. 27. Boston, MA: Kluwer Academic Publishers, 2000. p. 67–110.
   Google Scholar
• Zowe J, Kočvara M. Semidefinite programming. In: Ben-Tal A, Nemirovski A, editors. Modern optimization and its applications in engineering. Haifa (Israel): Technion; 2000.
   Google Scholar
• Yang SH. Semidefinite programming via image space analysis. J Indust Manag Optim. 2016;12:1187–1197.
   Web of Science ®Google Scholar
• Zaffaroni A. Degrees of efficiency and degrees of minimality. SIAM J Control Optim. 2003;42:1071–1086.
   Web of Science ®Google Scholar
• Göpfert A, Riahi H, Tammer C, et al. Variational methods in partially ordered spaces. New York: Springer; 2003.
   Google Scholar
• Bianchi M, Pini R. Sensitivity for parametric vector equilibria. Optimization. 2006;55:221–230.
   Web of Science ®Google Scholar