Abstract
In this paper, the concept of arcwise connected interval-valued function is introduced. Some optimality conditions and duality results are discussed for a nonlinear interval optimization problem with arcwise connected interval-valued functions.
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Ben-Tal, A., Nemirovski, A.: Robust optimization - methodology and applications. Math. Program. Ser B 92, 453–480 (2002)
Shapiro, A.: Stochastic programming approach to optimization under uncertainty. Math. Program. Ser B 112, 183–220 (2008)
Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia (2009)
Sengupta, A., Pal, T.K.: Fuzzy Preference Ordering of Interval Numbers in Decision Problems. Springer-Verlag, Berlin Heidelberg (2009)
Falk, J.E.: Exact solutions of inexact linear programs. Oper. Res. 24, 783–787 (1976)
Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154–1157 (1973)
Soyster, A.L.: Inexact linear programming with generalized resource sets. Eur. J. Oper. Res. 3, 316–321 (1979)
Thuente, D.J.: Duality theory for generalized linear programs with computational methods. Oper. Res. 28, 1005–1011 (1980)
Wu, H.C.: Duality theory in interval-valued linear programming problems. J. Optim. Theory Appl. 150, 298–316 (2011)
Wu, H.C.: Wolfe duality for interval-valued optimization. J. Optim. Theory Appl. 138, 497–509 (2008)
Wu, H.C: On interval-valued nonlinear programming problems. J. Math. Anal. Appl. 338, 299–316 (2008)
Zhou, H.C., Wang, Y.J.: Optimality condition and mixed duality for interval-valued optimization. Fuzzy Info. Eng. AISC 62, 1315–1323 (2009)
Sun, Y.H., Wang, L.S.: Duality theory for interval-valued programming. Adv. Sci. Lett. 7, 643–646 (2012)
Sun, Y.H., Wang, L.S.: Mond-Weir’s type duality for interval-valued programming. CSAE Proc. IEEE Int. Conf. Comput. Cci. Autom. Eng., 27–30 (2012)
Sun, Y.H., Xu, X.M., Wang, L.S.: Duality and saddle-point type optimality for interval-valued programming. Optim. Lett. 8, 1077–1091 (2014)
Zhang, J.K.: Optimality condition and Wolfe duality for invex interval-valued nonlinear programming problems. J. Appl. Math. Article ID 641345 (2013)
Li, L.F., Liu, S.Y., Zhang, J.K.: Univex interval-valued mapping with differentiability and its application in nonlinear programming. J. Appl. Math. Article ID 383692 (2013)
Preda, V.: Interval-valued optimization problems involving (α, ρ)−right upper-Dini-derivative functions. The Scientific Word Journal, Article ID 750910 (2014)
Jayswal, A., Stancu-Minasian, I.M., Ahmad, I.: On sufficiency and duality for a class of interval-valued programming problems. Appl. Math. Comput. 218(8), 4119–4127 (2011)
Jayswal, A., Stancu-Minasian, I.M., Banerjee, J., Stancu, A.M.: Sufficiency and duality for optimization problems involving interval-valued invex functions in parametric form. Oper. Res. 15(1), 137–161 (2015)
Avriel, M., Zang, I.: Generalized arcwise connected functions and characterization of local-global minimum properties. J. Optim. Theory Appl. 32, 407–425 (1980)
Rapcsák T.: Convex programming on Riemannian manifolds. System Modelling and Optimization Lecture Notes in Control and Information Sciences 84, 733–740 (1986)
Bhatia, D., Mehra, A.: Optimality conditions and duality involving arcwise connected and generalized arcwise connected functions. J. Optim. Theory Appl. 100, 181–194 (1999)
Zhang, Q.X.: Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions. J. Glob. optim. 45, 615–629 (2009)
Moore, R.: Interval analysis. Englewood Cliffs, Prentice-Hall (1966)
Hansen, W.G.E.: Global optimization using interval analysis. Marcel Dekker, New York (2004)
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The authors are very grateful to the anonymous referees for their valuable comments and suggestions which improved the original manuscript greatly.
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Wang, H., Zhang, R. Optimality conditions and duality for arcwise connected interval optimization problems. OPSEARCH 52, 870–883 (2015). https://doi.org/10.1007/s12597-015-0213-x
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DOI: https://doi.org/10.1007/s12597-015-0213-x