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In this paper, we consider nondifferentiable multiobjective fractional programming problems. A concept of generalized convexity, which is called (C, α, ρ, d)-convexity, is first discussed. Based on this generalized convexity, we obtain efficiency conditions for multiobjective fractional programming (MFP). Furthermore, we establish duality results for three types of dual problems of (MFP) and present the corresponding duality theorems.Keywords Multiobjective fractional programming problem · (C, α, ρ, d)-convexity · Efficiency conditions · Global efficient solution · Duality Several authors have been interested in generalization of convexity in connection with sufficiency and duality theorems in optimization problems (Hiriart-Urruty 1978;Kaul and Kaur 1985). It is possible to generalize the notion of convexity and extend the validity of theorems to larger classes of optimization problems. Therefore, several classes of generalized convex functions are introduced by researchers including Schmitendorf introduced a new class of generalized type I vector-valued functions, which are generalizations of type I function introduced by Hanson and Mond (1987), and proved duality theorems for differentiable multiobjective optimization problems A. Chinchuluun · P.M. Pardalos ( )

The dual 2I d -framelets in (H s (R d ), H −s (R d )), s > 0, were introduced by Han and Shen (Constr Approx 29 (3): 2009). In this paper, we systematically study the Bessel property of multiwavelet sequences in Sobolev spaces. The conditions for Bessel multiwavelet sequence in H −s (R d ) take great difference from those for Bessel wavelet sequence in this space. Precisely, the Bessel property of multiwavelet sequence in H −s (R d ) is not only related to multiwavelets themselves but also to the corresponding refinable function vector. We construct a class of Bessel M-refinable function vectors with M being an isotropic dilation matrix, which have high Sobolev smoothness, and of which the mask symbols have high sum rules. Based on the constructed Bessel refinable function vector, an explicit algorithm is given for dualhaving high vanishing moments. On the other hand, based on the dual multiframelets, an algorithm for dual M-multiframelets with symmetry is given. In Section 6, we give an example to illustrate the constructing procedures of dual multiframelets.

In this paper, we define new vector generalized convexity, namely nondifferentiable vector (G f , β f )-invexity, for a given locally Lipschitz vector function f . Basing on this new nondifferentiable vector generalized invexity, we have managed to deal with nondifferentiable nonlinear programming problems under some assumptions. Firstly, we present G-Karush-Kuhn-Tucker necessary optimality conditions for nonsmooth mathematical programming problems. With the new vector generalized invexity assumption, we also obtain G-Karush-Kuhn-Tucker sufficient optimality conditions for the same programming problems. Moreover, we establish duality results for this kind of multiobjective programming problems. In the end, a suitable example illustrates that the new optimality results are more useful for some class of optimization problems than the optimality conditions with invex functions. MSC: 90C26

In this paper, we give some new generalized convexities with the tool-right upper-Dini-derivative which is an extension of directional derivative. Next, we establish not only Karush-Kuhn-Tucker necessary but also sufficient optimality conditions for mathematical programming involving new generalized convex functions. In the end, weak, strong and converse duality results are proved to relate weak Pareto (efficient) solutions of the multi-objective programming problems (VP), (MVD) and (MWD).