Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

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E. Kiyani, S. M. Vaezpour, J. Tavakoli


In this paper, we consider, a new nonlinear scalarization function in vector spaces which is a generalization of the oriented distance function. Using the algebraic type of closure, which is called vector closure, we introduce the algebraic boundary of a set, without assuming any topology, in our context. Furthermore, some properties of this algebraic boundary set are given and present the concept of the oriented distance function via this set in the concept of vector optimization. We further investigate Q-proper efficiency in a real vector space, where Q is some nonempty (not necessarily convex) set. The necessary and sufficient conditions for Q-proper efficient solutions of nonconvex optimization problems are obtained via the scalarization technique. The scalarization technique relies on the use of two different scalarization functions, the oriented distance function and nonconvex separation function, which allow us to characterize the Q-proper efficiency in vector optimization with and without constraints.

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  1. M. Adán, V. Novo, Efficient and weak efficient points in vector optimization with generalized cone convexity, Appl. Math. Lett. 16 (2003), 221–225.
  2. M. Adán, V. Novo, Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness, Eur. J. Oper. Res. 149 (2003), 641–653.
  3. X.-H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior, J. Math. Anal. Appl. 307 (2005), 12–31.
  4. E. Hernández, B. Jiménez, V. Novo, Benson proper efficiency in set-valued optimization on real linear spaces, Lecture Notes in Economics and Mathematical Systems 563, Springer, Berlin, (2006) 45-59.
  5. E. Kiyani, M. Soleimani-damaneh, Approximate proper efficiency on real linear vector spaces, Pac. J. Optim. 10 (2013), 715-734.
  6. E. Kiyani, M. Soleimani-damaneh, Algebraic interior and separation on linear vector spaces: some comments, J. Optim. Theory Appl. 161 (2014), 994–998.
  7. E. Kiyani, S.M. Vaezpour, J. Tavakoli, Optimality conditions for approximate solution of set-valued optimization problems in real linear spaces, TWMS J. Appl. Eng. Math. 11 (2021), 395-407.
  8. M. Adán, V. Novo, Proper efficiency in vector optimization on real linear spaces, J. Optim. Theory Appl. 121 (2004), 515–540.
  9. Z.-A. Zhou, X.-M. Yang, J.-W. Peng, -Optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces, Optim. Lett. 8 (2014), 1047–1061.
  10. Z.-A. Zhou, X.-M. Yang, Scalarization of -super efficient solutions of set-valued optimization problems in real ordered linear spaces, J. Optim. Theory Appl. 162 (2014), 680–693.
  11. E. Hernández, B. Jiménez, V. Novo, Weak and proper efficiency in set-valued optimization on real linear spaces, J. Convex Anal. 14 (2007), 275–296.
  12. J. Jahn, Vector optimization, Theory, Applications, and Extensions, Springer, Berlin, 2011.
  13. J.H. Qiu, F. He, A general vectorial Ekeland’s variational principle with a P-distance, Acta. Math. Sin.-English Ser. 29 (2013), 1655–1678.
  14. J.-H. Qiu, A pre-order principle and set-valued Ekeland variational principle, J. Math. Anal. Appl. 419 (2014), 904–937.
  15. C. Gutiérrez, V. Novo, J.L. Ródenas-Pedregosa, T. Tanaka, Nonconvex separation functional in linear spaces with applications to vector equilibria, SIAM J. Optim. 26 (2016), 2677–2695.
  16. A.M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (1968), 618–630.
  17. H.W. Kuhn, A.W. Tucker, Nonlinear programming, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press. Berkeley. CA. (1951) 481-492.
  18. S. Wang, Z. Li, Scalarization and lagrange duality in multiobjective optimization, Optimization. 26 (1992) 315–324.
  19. S.J. Li, Y.D. Xu, S.K. Zhu, Nonlinear separation approach to constrained extremum problems, J. Optim. Theory Appl. 154 (2012), 842–856.
  20. J. Borwein, Proper efficient points for maximizations with respect to cones, SIAM J. Control Optim. 15 (1977), 57–63.
  21. T.X.D. Ha, Optimality conditions for several types of efficient solutions of set-valued optimization problems, in: P.M. Pardalos, T.M. Rassias, A.A. Khan (Eds.), Nonlinear Analysis and Variational Problems, Springer New York, New York, NY, 2010: pp. 305–324.
  22. S. Khoshkhabar-amiranloo, M. Soleimani-damaneh, Scalarization of set-valued optimization problems and variational inequalities in topological vector spaces, Nonlinear Analysis: Theory Meth. Appl. 75 (2012), 1429–1440.
  23. Z.A. Zhou, J.W. Peng, Scalarization of set-valued optimization problems with generalized cone subconvexlikeness in real ordered linear spaces, J. Optim. Theory Appl. 154 (2012), 830–841.
  24. C. Gerth, P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl. 67 (1990), 297–320.
  25. D.T. Luc, Theory of vector optimization. Lecture notes in economics and mathematical systems 319, Springer, Berlin, (1989).
  26. G.Y. Chen, X.X. Huang, X.G. Yang, Vector optimization: Set-valued and variational analysis, Springer-Verlag, Berlin, (2005).
  27. A. Göpfert, Chr. Tammer, C. Zălinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal.: Theory Meth. Appl. 39 (2000), 909–922.
  28. S.J. Li, X.Q. Yang, G.Y. Chen, Nonconvex vector optimization of set-valued mappings, J. Math. Anal. Appl. 283 (2003), 337–350.
  29. A.Göpfert, H. Riahi, C. Tammer, et al. Variational methods in Partially ordered spaces, Springer-Verlag, New York, (2003).
  30. E. Hernández, L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl. 325 (2007), 1–18.
  31. D. La Torre, N. Popovici, M. Rocca, Scalar characterizations of weakly cone-convex and weakly cone-quasiconvex functions, Nonlinear Anal.: Theory Meth. Appl. 72 (2010), 1909–1915.
  32. D. La Torre, N. Popovici, M. Rocca, A note on explicitly quasiconvex set-valued maps, J. Nonlinear Convex Anal. 12 (2011), 113-118.
  33. J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France, 60 (1979) 57-85.
  34. J. Jahn, Scalarization in multi objective optimization, in: P. Serafini (Ed.), Mathematics of Multi Objective Optimization, Springer Vienna, Vienna, 1985: pp. 45–88.
  35. G.P. Crespi, I. Ginchev, M. Rocca, First-order optimality conditions in set-valued optimization, Math. Meth. Oper. Res. 63 (2006), 87–106.
  36. G.P. Crespi, A. Guerraggio, M. Rocca, Well posedness in vector optimization problems and vector variational inequalities, J. Optim. Theory Appl. 132 (2007), 213–226.
  37. G.P. Crespi, M. Papalia, M. Rocca, Extended well-posedness of quasiconvex vector optimization problems, J. Optim. Theory Appl. 141 (2009), 285–297.
  38. G.P. Crespi, M. Papalia, M. Rocca, Extended well-posedness of vector optimization problems: the convex case, Taiwan. J. Math. 15 (2011), 1545-1559.
  39. Q. Qiu, X. Yang, Some properties of approximate solutions for vector optimization problem with set-valued functions, J Glob Optim. 47 (2010) 1–12.
  40. J.B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res. 4 (1979), 79–97.