Abstract
In this note, we show that for linear fractional vector optimization problems with bounded constraint sets there is no difference between the \(\epsilon \)-efficiency and the \(\epsilon \)-proper efficiency.
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Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)
Borwein, J.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)
Choo, E.U., Atkins, D.R.: Bicriteria linear fractional programming. J. Optim. Theory Appl. 36, 203–220 (1982)
Choo, E.U., Atkins, D.R.: Connectedness in multiple linear fractional programming. Manag. Sci. 29, 250–255 (1983)
Choo, E.U.: Proper efficiency and the linear fractional vector maximum problem. Oper. Res. 32, 216–220 (1984)
Chuong, T.D., Kim, D.S.: Approximate solutions of multiobjective optimization problems. Positivity 20, 187–207 (2016)
Geoffrion, A.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)
Gutiérrez, C., Huerga, L., Novo, V., Sama, M.: Limit behavior of approximate proper solutions in vector optimization. SIAM J. Optim. 29, 2677–2696 (2019)
Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. I. Fundamentals. Springer, Berlin (1993)
Hoa, T.N., Phuong, T.D., Yen, N.D.: Linear fractional vector optimization problems with many components in the solution sets. J. Ind. Manag. Optim. 1, 477–486 (2005)
Hong, Z., Piao, G.R., Kim, D.S.: On approximate solutions of nondifferentiable vector optimization problems with cone-convex objectives. Optim. Lett. 13, 891–906 (2019)
Hong, Z., Jiao, L.G., Kim, D.S.: On a class of nonsmooth fractional robust multi-objective optimization problems. Part I: Optimality conditions. Appl. Set-Valued Anal. Optim. 2, 109–121 (2020)
Huong, N.T.T., Yao, J.C., Yen, N.D.: Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets. J. Glob. Optim. 78, 545–562 (2020)
Huong, N.T.T., Yao, J.C., Yen, N.D.: New results on proper efficiency for a class of vector optimization problems. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2020.1712373
Isermann, H.: Proper efficiency and the linear vector maximum problem. Oper. Res. 22, 189–191 (1974)
Kim, D.S., Mordukhovich, B.S., Pham, T.S., Tuyen, N.V.: Existence of efficient and properly efficient solutions to problems of constrained vector optimization. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01532-y
Kutateladze, S.: Convex \(\epsilon \)-programming. Sov. Math. Dokl. 20, 391–393 (1979)
Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study. Springer, New York (2005)
Li, Z., Wang, S.: \(\epsilon \)-approximate solutions in multiobjective optimization. Optimization 44, 161–174 (1998)
Liu, J.C.: \(\epsilon \)-properly efficient solutions to nondifferentiable multiobjective programming problems. Appl. Math. Lett. 12, 109–113 (1999)
Long, X.J., Li, X.B., Zeng, J.: Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems. Optim. Lett. 7, 1847–1856 (2013)
Loridan, P.: \(\epsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)
Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)
Malivert, C.: Multicriteria fractional programming. In: Sofonea, M., Corvellec, J.N. (eds.) Proceedings of the 2nd Catalan Days on Applied Mathematics, pp. 189–198. Presses Universitaires de Perpinan (1995)
Mastroeni, G., Pappalardo, M., Raciti, F.: Some topics in vector optimization via image space analysis. J. Nonlinear Var. Anal. 4, 5–20 (2019)
Pareto, V.: Course d’Economie Politique. Rouge, Lausanne (1896)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, Orlando (1985)
Son, T.Q., Tuyen, N.V., Wen, C.F.: Optimality conditions for approximate Pareto solutions of a nonsmooth vector optimization problem with an infinite number of constraints. Acta Math. Vietnam 45, 435–448 (2020)
Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation and Application. Wiley, New York (1986)
Tigan, S.: Sur le problème de la programmation vectorielle fractionnaire. Rev. Anal. Numér. Théor. Approx. 4, 99–103 (1975)
Tuyen, N.V.: Approximate solutions of interval-valued optimization problems. Investigación Oper. 42, 223–237 (2021)
Yen, N.D.: Linear fractional and convex quadratic vector optimization problems. In: Ansari, Q.H., Yao, J.C. (eds.) Recent Developments in Vector Optimization, pp. 297–328. Springer, Berlin (2012)
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This research is funded by Hanoi Pedagogical University 2 under Grant Number HPU2.UT-2021.15.
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Tuyen, N.V. A note on approximate proper efficiency in linear fractional vector optimization. Optim Lett 16, 1835–1845 (2022). https://doi.org/10.1007/s11590-021-01806-0
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DOI: https://doi.org/10.1007/s11590-021-01806-0