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On gap functions for nonsmooth multiobjective optimization problems. (English) Zbl 1405.90115

It is known that, if at some feasible solution of a differentiable multiobjective problem the set-valued gap function introduced by G. Y. Chen et al. [Eur. J. Oper. Res. 111, No. 1, 142–151 (1998; Zbl 0944.90079)] contains zero, then that solution is efficient. In the present work, the authors prove that the converse statement is true if the solution is proper, and give a nonsmooth version of gap functions for convex problems. They also introduce a single-valued gap function and show that a feasible solution of a quasiconvex problem is efficient if and only if the gap function takes the value zero at that solution, provided that the so-called nonvanishing constraint qualification (NCQ) is satisfied. Note that the condition (NCQ) is not applicable to unconstrained single-criterion problems.

MSC:

90C29 Multi-objective and goal programming
49J52 Nonsmooth analysis

Citations:

Zbl 0944.90079
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Full Text: DOI

References:

[1] Altangerel, L., Boţ, R.I., Wanka, G.: Conjugate duality in vector optimization and some applications to the vector variational inequality. J. Math. Anal. Appl. 329, 1010-1035 (2007) · Zbl 1154.90613 · doi:10.1016/j.jmaa.2006.06.093
[2] Altangerel, L.: A duality approach to gap functions for variational inequalities and equilibrium problems, PhD Dissertation, Faculty of Mathematics, Chemnitz University of Technology, (2006) · Zbl 1162.49302
[3] Altangerel, L., Boţ, R.I., Wanka, G.: On gap functions for equilibrium problems via Fenchel duality. Pac. J. Optim. 2, 667-678 (2006) · Zbl 1103.49016
[4] Altangerel, L., Boţ, R.I., Wanka, G.: On the construction of gap functions for variational inequalities via conjugate duality. Asia Pac. J. Oper. Res. 24, 353-371 (2007) · Zbl 1141.49303 · doi:10.1142/S0217595907001309
[5] Auslender, A.: Optimisation: Méthods Numériques. Masson, Paris (1976) · Zbl 0326.90057
[6] Bagirov, A., Karmitsa, N., Mäkelä, M.M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Cham, Heidelberg (2014) · Zbl 1312.90053 · doi:10.1007/978-3-319-08114-4
[7] Benson, H.P.: Existence of efficient solutions for vector maximization problems. J. Optim. Theory Appl. 26, 569-580 (1978) · Zbl 0373.90085 · doi:10.1007/BF00933152
[8] Benson, H.P.: Optimization over the efficient set. J. Math. Anal. Appl. 98, 562-580 (1984) · Zbl 0534.90077 · doi:10.1016/0022-247X(84)90269-5
[9] Boţ, R.I., Grad, S.M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin, Heidelberg · Zbl 1177.90355
[10] Chen, C.Y., Goh, C.J., Yang, X.Q.: The gap function of a convex multicriteria optimization problem. Eur. J. Oper. Res. 111, 142-151 (1998) · Zbl 0944.90079 · doi:10.1016/S0377-2217(97)00300-7
[11] Clarke, F.H.: Functional Analysis. Calculus of Variations and Optimal Control. Springer, London (2013) · Zbl 1277.49001 · doi:10.1007/978-1-4471-4820-3
[12] Daniilidis, A., Hadjisavvas, N.: Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions. J. Optim. Theory Appl. 102, 525-536 (1999) · Zbl 1010.49013 · doi:10.1023/A:1022693822102
[13] Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005) · Zbl 1132.90001
[14] Flores-Bazán, F., Mastroeni, G.: Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications. SIAM J. Optim. 25, 647-676 (2015) · Zbl 1398.90126 · doi:10.1137/13094606X
[15] Geoffrion, A.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618-630 (1968) · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1
[16] Hearn, D.W.: The gap function of a convex program. Oper. Res. Lett. 1, 67-71 (1982) · Zbl 0486.90070 · doi:10.1016/0167-6377(82)90049-9
[17] Hurwicz, L.; Arrow, KJ (ed.); Hurwicz, L. (ed.); Uzawa, H. (ed.), Programming in linear spaces, 38-102 (1958), Stanford
[18] Jourani, A.: Metric regularity and second-order necessary optimality conditions for minimization problems under inclusion constraints. J. Glob. Optim. 81, 97-120 (1994) · Zbl 0808.49023
[19] Kuhn, H., Tucker, A.: Nonlinear programing. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481-492. University of California Press, Berkeley, California (1951) · Zbl 1010.49013
[20] Lasserre, J.B.: On representation of the feasible set in convex optimization. Optim. Lett. 4, 1-5 (2010) · Zbl 1180.90237 · doi:10.1007/s11590-009-0153-6
[21] Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27, 411-426 (2003) · Zbl 1061.90112 · doi:10.1023/A:1026050425030
[22] Mirzaee, H., Soleimani-damaneh, M.: Optimality, duality and gap function for quasi variational inequality problems. ESAIM Control Optim. Calc. Var. 23, 297-308 (2017) · Zbl 1365.49013 · doi:10.1051/cocv/2015053
[23] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)
[24] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401 · doi:10.1515/9781400873173
[25] Sarabi, M.E., Soleimani-damaneh, M.: Revisiting the function of a multicriteria optimization problem. Int. J. Comput. Math. 86, 860-863 (2009) · Zbl 1173.90538 · doi:10.1080/00207160701713607
[26] Soleimani-damaneh, M.: Characterization of nonsmooth quasiconvex and pseudoconvex functions. J. Math. Anal. Appl. 330, 2168-2176 (2007) · Zbl 1162.49302 · doi:10.1016/j.jmaa.2006.08.033
[27] Soleimani-damaneh, M.: The gap function for optimization problems in Banach spaces. Nonlinear Anal. 69, 716-723 (2008) · Zbl 1211.90218 · doi:10.1016/j.na.2007.06.008
[28] Soleimani-damaneh, M.: Nonsmooth optimization using Mordukhovich’s subdifferential. SIAM J. Control Optim. 48, 3403-3432 (2010) · Zbl 1204.47095 · doi:10.1137/070710664
[29] Yamamoto, Y.: Optimization over the efficient set: overview. J. Glob. Optim. 22, 285-317 (2002) · Zbl 1045.90061 · doi:10.1023/A:1013875600711
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