Abstract
In this paper, a generalization of convexity, namely G-invexity, is considered in the case of nonlinear multiobjective programming problems where the functions constituting vector optimization problems are differentiable. The modified Karush-Kuhn-Tucker necessary optimality conditions for a certain class of multiobjective programming problems are established. To prove this result, the Kuhn-Tucker constraint qualification and the definition of the Bouligand tangent cone for a set are used. The assumptions on (weak) Pareto optimal solutions are relaxed by means of vector-valued G-invex functions.
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References
Antczak T. (2001). (p, r)-invex sets and functions. J. Math. Anal. Applic. 263: 355–379
Antczak T. (2004). (p, r)-invexity in multiobjective programming. Eur. J. Oper. Res. 152: 72–87
Antczak T. (2005). The notion of V-r-invexity in differentiable multiobjective programming. J. Appl. Anal. 11: 63–79
Antczak T. (2007). New optimality conditions and duality results of G-type in differentiable mathematical programming. Nonlinear Anal. 66: 1617–1632
Batista dos Santos, L., Osuna-Gomez, R., Rojas-Medar, M.A., Rufian-Lizana, A.: Invexity generalized and weakly efficient solutions for some vectorial optimization problem in Banach spaces. Numer. Funct. Anal. Optim. (2004)
Bazaraa M.S., Sherali H.D. and Shetty C.M. (1991). Nonlinear programming: theory and algorithms. Wiley, New York
Ben-Israel A. and Mond B. (1986). What is invexity? J. Aust. Math. Soc. Ser. B 28: 1–9
Craven B.D. (1981). Invex functions and constrained local minima. Bulletin of the Aust. Math. Soc. 24: 357–366
Craven B.D. and Glover B.M. (1985). Invex functions and duality. J. Aust. Math. Soc. Ser .A 39: 1–20
Geoffrion M.A. (1968). Proper efficiency and the theory of vector maximization. J. Math. Anal. Applic. 22: 613–630
Giorgi, G., Guerraggio, A.: The notion of invexity in vector optimization: smooth and nonsmooth case. In: Crouzeix, J.P. et al. (eds.), Generalized Convexity, Generalized Monotonicity. Kluwer Academic Publishers (1998)
Hanson M.A. (1981). On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Applic. 80: 545–550
Hanson M.A. and Mond B. (1987). Necessary and sufficient conditions in constrained optimization. Math. Programming 37: 51–58
Jeyakumar V. and Mond B. (1992). On generalized convex mathematical programming. J. Aust. Mathe. Soc. Ser. B 34: 43–53
Kanniappan P. (1983). Necessary conditions for optimality of nondifferentiable convex multiobjective programming. J. Optim. Theory Applic. 40: 167–174
Kaul R.N., Suneja S.K. and Srivastava M.K. (1994). Optimality criteria and duality in multiple objective optimization involving generalized invexity. J. Optim. Theory Applic. 80: 465–482
Lin J.G. (1976). Maximal vectors and multi-objective optimization. J. Optim. Theory Applic. 18: 41–64
Mangasarian O.L. (1969). Nonlinear programming. McGraw-Hill, New York
Martin D.H. (1985). The essence of invexity. J. Optim. Theory Applic. 47: 65–76
Osuna-Gómez R., Rufián-Lizana A. and Ruiz-Canales P. (1998). Invex functions and generalized convexity in multiobjective programming. J. Optim. Theory Applic. 98: 651–661
Pareto, V.: Cours de economie politique. Rouge, Lausanne, Switzerland (1896)
Singh C. (1987). Optimality conditions in multiobjective differentiable programming. J. Optim. Theory Applic. 53: 115–123
Varaiya P.P. (1972). Notes on optimization. Van Nostrand Reinhold Company, New York, New York
Weir T. (1987). Proper efficiency and duality for vector valued optimization problems. J. Aust. Math. Soc. Ser. A 43: 24–34
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Antczak, T. On G-invex multiobjective programming. Part I. Optimality. J Glob Optim 43, 97–109 (2009). https://doi.org/10.1007/s10898-008-9299-5
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DOI: https://doi.org/10.1007/s10898-008-9299-5