This bibliography of fractional programming is a continuation of eight previous bibliographies by the author (Pure Appl. Math. Sci. (India), Vol. XIII, No. 1–2, 35–69, March (1981); ibid. Vol. XVII...

(2017). A eighth bibliography of fractional programming. Optimization: Vol. 66, No. 3, pp. 439-470.

In this article, we study nonsmooth interval-valued multiobjective programming problem and generalized Stampacchia vector variational inequality with its weak form for interval-valued functions. Using the tools of Mordukhovich subdifferential, we define some new classes of generalized approximate LU-convex functions. These functions are then employed to establish the relations between the solutions of generalized vector variational inequalities and the approximate LU-efficient solutions of the nonsmooth interval-valued multiobjective programming problem. Moreover, we identify the Kuhn–Tucker vector critical points of the considered nonsmooth interval-valued multiobjective programming problem. Under suitable constraint qualification, we establish the equivalence among local approximate LU-efficient points, Kuhn–Tucker vector critical points and the solutions of generalized Stampacchia vector variational inequalities. The results of this paper extend and sharpen the corresponding results of [Giannessi F. On Minty variational principle. In: Giannessi F, Komlósi S, Rapcśak T. editors. New trends in mathematical programming. Dordrecht: Kluwer Academic Publishers; 1997. p. 93–99], [Golestani M, Sadeghi H, Tavan Y. Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi-efficiency. J Optim Theory Appl. 2018;179(3):896–916], [Lee GM, Lee KB. Vector variational inequalities for nondifferentiable convex vector optimization problems. J Global Optim. 2005;32(4):597–612], [Mishra SK, Upadhyay BB. Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity. 2013;17(4):1071–1083], [Upadhyay BB, Mohapatra RN, Mishra SK. On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv Nonlinear Var Inequal. 2017;20(2):1–12], [Zhang J, Zheng Q,Ma X, Li L. Relationships between interval-valued vector optimization problems and vector variational inequalities. Fuzzy Optim Decis Mak. 2016;15(1):33–55] for nonsmooth interval-valued multiobjective programming problem by using the powerful tool of Mordukhovich subdifferential.

This article deals with multiobjective fractional programming problems with equilibrium constraints in the setting of Hadamard manifolds (abbreviated as MFPPEC). The generalized Guignard constraint qualification (abbreviated as GGCQ) for MFPPEC is presented. Furthermore, the Karush–Kuhn–Tucker (abbreviated as KKT) type necessary criteria of Pareto efficiency for MFPPEC are derived using GGCQ. Sufficient criteria of Pareto efficiency for MFPPEC are deduced under some geodesic convexity hypotheses. Subsequently, Mond–Weir and Wolfe type dual models related to MFPPEC are formulated. The weak, strong, and strict converse duality results are derived relating MFPPEC and the respective dual models. Suitable nontrivial examples have been furnished to demonstrate the significance of the results established in this article. The results derived in the article extend and generalize several notable results previously existing in the literature. To the best of our knowledge, optimality conditions and duality for MFPPEC have not yet been studied in the framework of manifolds.