Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with a bounded constraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).
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Benoist, J.: Connectedness of the efficient set for strictly quasiconcave sets. J. Optim. Theory Appl. 96, 627–654 (1998)
Benoist, J.: Contractibility of the efficient set in strictly quasiconcave vector maximization. J. Optim. Theory Appl. 110, 325–336 (2001)
Benson, B.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)
Bochnak, R., Coste, M., Roy, M.F.: Real Algebraic Geometry. Spinger, Berlin (1998)
Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)
Chew, K.L., Choo, E.U.: Pseudolinearity and efficiency. Math. Program. 28, 226–239 (1984)
Choo, E.U.: Proper efficiency and the linear fractional vector maximum problem. Oper. Res. 32, 216–220 (1984)
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)
Crespi, G.P.: Proper efficiency and vector variational inequalities. J. Inform. Optim. Sci. 23, 49–62 (2002)
Crespi, G.P., Ginchev, I., Rocca, M.: Existence of solutions and star-shapedness in Minty variational inequalities. J. Global Optim. 32, 485–494 (2005)
Crespi, G.P., Ginchev, I., Rocca, M.: Increasing-along-rays property for vector functions. J. Nonlinear Convex Anal. 7, 39–50 (2006)
Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 613–630 (1968)
Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht (2000)
Guerraggio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)
Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)
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)
Hoa, T.N., Phuong, T.D., Yen, N.D.: On the parametric affine variational inequality approach to linear fractional vector optimization problems. Vietnam J. Math. 33, 477–489 (2005)
Hoa, T.N., Huy, N.Q., Phuong, T.D., Yen, N.D.: Unbounded components in the solution sets of strictly quasiconcave vector maximization problems. J. Glob. Optim. 37, 1–10 (2007)
Huong, N.T.T., Hoa, T.N., Phuong, T.D., Yen, N.D.: A property of bicriteria affine vector variational inequalities. Appl. Anal. 10, 1867–1879 (2012)
Huong, N.T.T., Yao, J.-C., Yen, N.D.: Connectedness structure of the solution sets of vector variational inequalities. Optimization 66, 889–901 (2017)
Huy, N.Q., Yen, N.D.: Remarks on a conjecture of. J. Benoist. Nonlinear Anal. Forum 9, 109–117 (2004)
Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study Series: Nonconvex Optimization and its Applications, vol. 78. Springer, New York (2005)
Luc, D.T.: Multiobjective Linear Programming: An Introduction. Springer, Berlin (2016)
Malivert C.: Multicriteria fractional programming. In: Sofonea, M., Corvellec, J.N. (eds.) Proceedings of the 2nd Catalan Days on Applied Mathematics. Presses Universitaires de Perpinan, pp. 189–198 (1995)
Robinson, S.M.: Generalized equations and their solutions. Part I: Basic theory. Math. Program. Study 10, 128–141 (1979)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rubinov, A.M., Glover, B.M.: Increasing convex-along-rays functions with applications to global optimization. J. Optim. Theory Appl. 102, 615–642 (1999)
Stancu-Minasian, I.M.: A ninth bibliography of fractional programming. Optimization 68, 2123–2167 (2019)
Steuer, R.E.: Multiple Criteria Optimization: Theory Computation and Application. Wiley, New York (1986)
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)
Yen N.D., Phuong T.D.: Connectedness and stability of the solution set in linear fractional vector optimization problems. 38, 479–489. https://link.springer.com/chapter/10.1007/978-1-4613-0299-5_29. (2000)
Yen, N.D., Yang, X.Q.: Affine variational inequalities on normed spaces. J. Optim. Theory Appl. 178, 36–55 (2018)
Yen, N.D., Yao, J.-C.: Monotone affine vector variational inequalities. Optimization 60, 53–68 (2011)
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The authors were supported by National Foundation for Science and Technology Development (Vietnam) under Grant No. 101.01-2018.306, Le Quy Don Technical University (Vietnam), Grant MOST 105-2115-M-039-002-MY3 (Taiwan), and the Vietnam Institute for Advanced Study in Mathematics (VIASM, Vietnam).
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Huong, N.T.T., Yao, JC. & Yen, N.D. Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets. J Glob Optim 78, 545–562 (2020). https://doi.org/10.1007/s10898-020-00927-7
- Linear fractional vector optimization
- Efficient solution
- Gain-to-loss ratio
- Geoffrion’s properly efficient solution
- Unbounded constraint set
- Direction of recession
- Regularity condition
Mathematics Subject Classification