External and internal stability in set optimization

References

• Chen GY, Huang XX, Yang XQ. Vector optimization: set-valued and variational analysis. Berlin: Springer-Verlag; 2005. (Lecture notes in economics and mathematical system; vol. 541).
   Google Scholar
• Aubin J-P, Cellina A. Differential inclusions: set-valued maps and viability theory. Berlin: Springer-Verlag; 1984. (Grundlehren Math Wiss).
   Google Scholar
• Klein E, Thompson AC. Theory of correspondences: including applications to mathematical economics. New York (NY): Wiley; 1984.
   Google Scholar
• Khan AA, Tammer C, Zlinescu C. Set-valued optimization: an introduction with applications. Berlin: Springer; 2015.
   Google Scholar
• Aubin J-P, Ekeland I. Applied nonlinear analysis. New York (NY): Wiley; 1984.
   Google Scholar
• Corley HW. Existence and Lagrangian duality for maximizations of set-valued functions. J Optim Theory Appl. 1987;54:489–501.
   Web of Science ®Google Scholar
• Li SJ, Yang XQ, Chen GY. Nonconvex vector optimization of set-valued mappings. J Math Anal Appl. 2003;283:337–350.
   Web of Science ®Google Scholar
• Kuroiwa D. On set-valued optimization. Nonlinear Anal. 2001;47:1395–1400.
   Web of Science ®Google Scholar
• Kuroiwa D. The natural criteria in set-valued optimization. $\text{S}\overline{\text{u}}\text{rikaisekikenky}\overline{\text{u}}\text{sho}$ $\text{K}\overline{\text{o}}\text{ky}\overline{\text{u}}\text{roku}$. 1999;1031:85–90.
   Google Scholar
• Hernández E, Rodríguez-Marín L. Nonconvex scalarization in set optimization with set-valued maps. J Math Anal Appl. 2007;325:1–18.
   Web of Science ®Google Scholar
• Dhingra M, Lalitha CS. Well-setness and scalarization in set optimization. Optim Lett. 2016;10:1657–1667.
   Web of Science ®Google Scholar
• Zhang WY, Li SJ, Teo KL. Well-posedness for set optimization problems. Nonlinear Anal. 2008;71:3768–3788.
   Web of Science ®Google Scholar
• Lalitha CS, Dhingra M. Approximate Lagrangian duality and saddle point optimality in set optimization. RAIRO-Oper Res. 2017;51:819–831.
   Web of Science ®Google Scholar
• Han Yu, Huang NJ. Well-posedness and stability of solutions for set optimization problems. Optimization. 2017;66:17–33.
   Web of Science ®Google Scholar
• Sawaragi Y, Nakayama H, Tanino T. Theory of multiobjective optimization. New York (NY): Academic Press; 1985.
   Google Scholar
• Xu YD, Li SJ. Continuity of solution set mappings to a parametric set optimization problem. Optim Lett. 2014;8:2315–2327.
   Web of Science ®Google Scholar
• Xu YD, Li SJ. On the solution continuity of parametric set optimization problems. Math Methods Oper Res. 2016;84:223–237.
   Web of Science ®Google Scholar
• Babuška I, Hlaváček I, Chleboun J. Uncertain input data problems and the worst scenario method. Amsterdam: Elsevier; 2004. (North-Holland series in applied mathematics and mechanics; vol. 46).
   Google Scholar
• Eslami M. Theory of sensitivity in dynamical systems: an introduction. Berlin: Springer-Verlag; 1994.
   Google Scholar
• Fiacco AV. Introduction to sensitivity and stability analysis in nonlinear programming. New York (NY): Academic Press; 1983.
   Google Scholar
• Wierzbicki AP. Models and sensitivity of control systems. Amsterdam: Elsevier-WNT; 1984.
   Google Scholar
• Attouch H, Riahi H. Stability results for Ekeland's ϵ-variational principal and cone extremal solution. Math Oper Res. 1993;18:173–201.
   Web of Science ®Google Scholar
• Luc DT, Lucchetti R, Malivert C. Convergence of efficient sets. Set-Valued Var Anal. 1994;2:207–218.
   Google Scholar
• Huang XX. Stability in vector-valued and set-valued optimization. Math Methods Oper Res. 2000;52:185–193.
   Web of Science ®Google Scholar
• Lalitha CS, Chatterjee P. Stability for properly quasiconvex vector optimization problem. J Optim Theory Appl. 2012;155:492–506.
   Web of Science ®Google Scholar
• Lalitha CS, Chatterjee P. Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities. J Optim Theory Appl. 2012;155:941–961.
   Web of Science ®Google Scholar
• Li XB, Lin Z, Peng ZY. Convergence for vector optimization problems with variable ordering structure. Optimization. 2016;65:1615–1627.
   Web of Science ®Google Scholar
• Li XB, Wang QL, Lin Z. Stability of set-valued optimization problems with naturally quasi-functions. J Optim Theory Appl. 2016;168:850–863.
   Web of Science ®Google Scholar
• Gutiérrez C, Miglierina E, Mohlo E, et al. Convergence of solutions of a set optimization problem in the image space. J Optim Theory Appl. 2016;170:358–371.
   Web of Science ®Google Scholar
• Kuroiwa D. On duality of set-valued optimization – research on nonlinear analysis and convex analysis. $\text{S}\overline{\text{u}}\text{rikaisekikenky}\overline{\text{u}}\text{sho}$ $\text{K}\overline{\text{o}}\text{ky}\overline{\text{u}}\text{roku}$. 1998;1071:12–16.
   Google Scholar
• Kuroiwa D. Some duality theorems of set-valued optimization with natural criteria. Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis. River Edge (NJ): World Scientific; 1999. p. 221–228.
   Google Scholar
• Kuratowski K. Topology. Vol. 2. New York (NY): Academic Press; 1968.
   Google Scholar
• Luc DT. Theory of vector optimization. Berlin: Springer-Verlag; 1989. (Lecture notes in economics and mathematical systems; vol. 319).
   Google Scholar
• Karuna, Lalitha CS. Continuity of approximate weak efficient solution set map in parametric set optimization. J Nonlinear Convex Anal. 2018;19:1247–1262.
   Web of Science ®Google Scholar
• Luc DT. On the domination property in vector optimization. J Optim Theory Appl. 1984;2:327–330.
   Web of Science ®Google Scholar
• Hernández E, Rodríguez-Marín L. Existence theorems for set optimization problems. Nonlinear Anal. 2007;67:1726–1736.
   Web of Science ®Google Scholar