Abstract
In this work we introduce two approximate duality approaches for vector optimization problems. The first one by means of approximate solutions of a scalar Lagrangian, and the second one by considering \((C,\varepsilon )\)-proper efficient solutions of a recently introduced set-valued vector Lagrangian. In both approaches we obtain weak and strong duality results for \((C,\varepsilon )\)-proper efficient solutions of the primal problem, under generalized convexity assumptions. Due to the suitable limit behaviour of the \((C,\varepsilon )\)-proper efficient solutions when the error \(\varepsilon \) tends to zero, the obtained duality results extend and improve several others in the literature.
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References
Boţ, R.I., Grad, S.-M.: Duality for vector optimization problems via a general scalarization. Optimization 60, 1269–1290 (2011)
Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin (2009)
Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (I). Optimization 53, 281–300 (2004)
Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (II). Optimization 53, 301–324 (2004)
Dutta, J., Vetrivel, V.: On approximate minima in vector optimization. Numer. Funct. Anal. Optim. 22, 845–859 (2001)
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and \(\varepsilon \)-subdifferentials in vector optimization: basic properties and limit behaviour. Nonlinear Anal. 79, 52–67 (2013)
Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and \(\varepsilon \)-subdifferentials in vector optimization: Moreau–Rockafellar type theorems. J. Convex Anal. 21, 857–886 (2014)
Gutiérrez, C., Huerga, L., Novo, V.: Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems. J. Math. Anal. Appl. 389, 1046–1058 (2012)
Gutiérrez, C., Jiménez, B., Novo, V.: Multiplier rules and saddle-point theorems for Helbig’s approximate solutions in convex Pareto problems. J. Glob. Optim. 32, 367–383 (2005)
Gutiérrez, C., Jiménez, B., Novo, V.: On approximate efficiency in multiobjective programming. Math. Methods Oper. Res. 64, 165–185 (2006)
Gutiérrez, C., Jiménez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17, 688–710 (2006)
Gutiérrez, C., Jiménez, B., Novo, V.: A generic approach to approximate efficiency and applications to vector optimization with set-valued maps. J. Glob. Optim. 49, 313–342 (2011)
Jahn, J.: Duality in vector optimization. Math. Program. 25, 343–353 (1983)
Jahn, J.: Vector Optimization. Theory, Applications and Extensions. Springer, Berlin (2011)
Jia, J.-H., Li, Z.-F.: \(\varepsilon \)-Conjugate maps and \(\varepsilon \)-conjugate duality in vector optimization with set-valued maps. Optimization 57, 621–633 (2008)
Kutateladze, S.S.: Convex \(\varepsilon \)-programming. Soviet Math. Dokl. 20, 391–393 (1979)
Lemaire, B.: Approximation in multiobjective optimization. J. Glob. Optim. 2, 117–132 (1992)
Li, Z.F.: Benson proper efficiency in the vector optimization of set-valued maps. J. Optim. Theory Appl. 98, 623–649 (1998)
Luc, D.T.: On duality theory in multiobjective programming. J. Optim. Theory Appl. 43, 557–582 (1984)
El Maghri, M.: Pareto-Fenchel \(\varepsilon \)-subdifferential sum rule and \(\varepsilon \)-efficiency. Optim. Lett. 6, 763–781 (2012)
El Maghri, M.: (\(\varepsilon \)-)Efficiency in difference vector optimization. J. Glob. Optim. 61, 803–812 (2015)
Qiu, J.H.: Dual characterization and scalarization for Benson proper efficiency. SIAM J. Optim. 19, 144–162 (2008)
Rong, W.D., Wu, Y.N.: \(\varepsilon \)-Weak minimal solutions of vector optimization problems with set-valued maps. J. Optim. Theory Appl. 106, 569–579 (2000)
Sach, P.H., Tuan, L.A., Minh, N.B.: Approximate duality for vector quasi-equilibrium problems and applications. Nonlinear Anal. 72, 3994–4004 (2010)
Son, T.Q., Kim, D.S.: \(\varepsilon \)-Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints. J. Glob. Optim. 57, 447–465 (2013)
Tanino, T., Sawaragi, Y.: Duality theory in multiobjective programming. J. Optim. Theory Appl. 27, 509–529 (1979)
Vályi, I.: Approximate saddle-point theorems in vector optimization. J. Optim. Theory Appl. 55, 435–448 (1987)
Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)
Acknowledgments
This work was partially supported by Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942. The authors are very grateful to the anonymous referees for their helpful comments and suggestions.
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L. Huerga: Researcher of Spanish FPI Fellowship Programme (BES-2010-033742).
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Gutiérrez, C., Huerga, L., Novo, V. et al. Duality related to approximate proper solutions of vector optimization problems. J Glob Optim 64, 117–139 (2016). https://doi.org/10.1007/s10898-015-0366-4
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DOI: https://doi.org/10.1007/s10898-015-0366-4
Keywords
- Vector optimization
- Approximate duality
- Proper \(\varepsilon \)-efficiency
- Nearly cone-subconvexlikeness
- Linear scalarization