Semicontinuity conditions for set-valued equilibrium problems and applications to set optimization problems

• Nguyen Huu Danh ^[1]
  1. [1] Taydo University, Cantho, Vietnam
• Localización: Top, ISSN-e 1863-8279, ISSN 1134-5764, Vol. 34, Nº. 1, 2026, págs. 24-39
• Idioma: inglés
• DOI: 10.1007/s11750-025-00701-6
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we examine parametric set-valued equilibrium problems. We begin by introducing a quasiconvexity property for set-valued maps and exploring its relationship with existing concepts. Next, we analyze the semicontinuity and continuity of approximate solution maps for these problems, without assuming the solid condition of the ordered cone and the compact values of the objective map. Finally, we demonstrate an application of the main results to set optimization problems involving a possibly less order relation.

• Referencias bibliográficas
  • Alleche B, Rădulescu VD (2017) Further on set-valued equilibrium problems and applications to Browder variational inclusions. J Optim Theory...
  • Alonso M, Rodríguez-Marín L (2009) Optimality conditions for set-valued maps with set optimization. Nonlinear Anal 70(9):3057–3064
  • Anh LQ, Duoc PT, Tam TN (2020a) On the stability of approximate solutions to set-valued equilibrium problems. Optimization 69(7–8):1583–1599
  • Anh LQ, Danh NH, Tam TN (2020b) Continuity of solution maps to parametric set optimization problems via parametric equilibrium problems. Acta...
  • Anh LQ, Duoc PT, Tam TN, Thang NC (2021) Stability analysis for set-valued equilibrium problems with applications to Browder variational inclusions....
  • Anh LQ, Danh NH, Duoc PT (2024) Hausdorff continuity of solution maps to equilibrium problems via the oriented distance function. Optim Lett...
  • Anh LQ, Duoc PT, Linh HM (2024) Scalar representations and Hausdorff continuity of solution mappings to parametric set optimization problems...
  • Ansari QH, Sharma PK, Hussain N (2024) Semi-continuity of the solution maps of set-valued equilibrium problems with equilibrium constraints....
  • Aubin JP, Frankowska H (1990) Set-valued analysis. Birkhäuser Boston. Inc., Boston
  • Bigi G, Castellani M, Pappalardo M, Passacantando M (2019) Nonlinear programming techniques for equilibria. Springer, Cham
  • Durea M (2007) On the existence and stability of approximate solutions of perturbed vector equilibrium problems. J Math Anal Appl 333(2):1165–1179
  • Durea M, Florea E-A (2024a) Cone-compactness of a set and applications to set-equilibrium problems. J Optim Theory Appl 200(3):1286–1308
  • Durea M, Florea E-A (2024b) Subdifferential calculus and ideal solutions for set optimization problems. J Nonlinear Var Anal 8:533–547
  • Eslamizadeh L, Naraghirad E (2020) Existence of solutions of set-valued equilibrium problems in topological vector spaces with applications....
  • Geering HP (2007) Optimal control with engineering applications, vol 113. Springer, Berlin
  • Goeleven D (2017) Complementarity and variational inequalities in electronics. Academic Press, London
  • Göpfert A, Riahi H, Tammer C, Zalinescu C (2003) Variational methods in partially ordered spaces. Springer, New York
  • Gutiérrez C, Miglierina E, Molho E, Novo V (2012) Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal 75(4):1822–1833
  • Han Y (2019) Nonlinear scalarizing functions in set optimization problems. Optimization 68(9):1685–1718
  • Han Y (2025) Cone sequential compactness of a set and an application to set optimization problems. J Optim Theory Appl 206:59
  • Han Y, Huang NJ (2016) Some characterizations of the approximate solutions to generalized vector equilibrium problems. J Ind Manag Optim 12(3):1135–1151
  • Han Y, Huang NJ (2017) Well-posedness and stability of solutions for set optimization problems. Optimization 66(1):17–33
  • Han W, Yu G (2022) Scalarization and semicontinuity of approximate solutions to set optimization problems. Appl Set-Valued Anal Optim 4(2):239–250
  • Hernández E, Rodríguez-Marín L (2007) Existence theorems for set optimization problems. Nonlinear Anal 67(6):1726–1736
  • Hernández E, Rodríguez-Marín L (2007) Lagrangian duality in set-valued optimization. J Optim Theory Appl 134:119–134
  • Hu S, Papageorgiou SHN (1997) Handbook of multivalued analysis. Volume I: theory. Mathematics and its applications. Kluwer, Boston
  • Jafari S, Farajzadeh AP, Moradi S, Khanh PQ (2017) Existence results for-quasimonotone equilibrium problems in convex metric spaces. Optimization...
  • Jahn J, Ha TXD (2011) New order relations in set optimization. J Optim Theory Appl 148(2):209–236
  • Kassay G, Rădulescu V (2018) Equilibrium problems and applications. Academic Press, London
  • Khan AA, Tammer C, Zălinescu C (2015) Set-valued optimization: an introduction with applications. Springer, Berlin
  • Khoshkhabar-amiranloo S, Khorram E (2015) Pointwise well-posedness and scalarization in set optimization. Math Methods Oper Res 82(2):195–210
  • Kinderlehrer D, Stampacchia G (2000) An introduction to variational inequalities and their applications. SIAM, Philadelphia
  • Kuroiwa D, Tanaka T, Ha TXD (1997) On cone convexity of set-valued maps. Nonlinear Anal 30(3):1487–1496
  • Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman and Hall/CRC, Boca Raton
  • Li XB, Lin Z, Wang QL (2016) Stability of approximate solution mappings for generalized Ky Fan inequality. TOP 24(1):196–205
  • Mordukhovich BS (2018) Variational analysis and applications. Springer, Berlin
  • Seto K, Kuroiwa D, Popovici N (2018) A systematization of convexity and quasiconvexity concepts for set-valued maps, defined by l-type and...
  • Shehu Y, Dong QL, Liu L, Yao JC (2023) Alternated inertial subgradient extragradient method for equilibrium problems. TOP 31:1–30
  • TN Tam (2022) On Hölder continuity of solution maps to parametric vector Ky Fan inequalities. TOP 30(1):77–94
  • Tung NM (2022) Karush-Kuhn-Tucker multiplier rules for efficient solutions of set-valued equilibrium problem with constraints. Bull Iran Math...
  • Xu Y, Li S (2014) Continuity of the solution set mappings to a parametric set optimization problem. Optim Lett 8(8):2315–2327
  • Xu Y, Li S (2016) On the solution continuity of parametric set optimization problems. Math Meth Oper Res 84(1):223–237
  • Zhao X, Ghosh D, Qin X, Tammer C, Yao JC (2025) On the convergence analysis of a proximal gradient method for multiobjective optimization....