R. Osuna-Gómez

Invex functions and generalized convexity in multiobjective programming

Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn–Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn–Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.

Relationships between vector variational-like inequality and optimization problems

Abstract In this paper we will establish the relationships between vector variational-like inequality and optimization problems. We will be able to identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality problem, under conditions of pseudo invexity. These conditions are more general those existing in the literature.

Generalized invex monotonicity

Abstract In this paper the generalized invex monotone functions are defined as an extension of monotone functions. A series of sufficient and necessary conditions are also given that relate the generalized invexity of the function θ with the generalized invex monotonicity of its gradient function ∇θ. This new class of functions will be important in order to characterize the solutions of the Variational-like Inequality Problem and Mathematical Programming Problem.

KT-invex control problem

Abstract In this paper, we introduce a new condition on functions of a control problem, for which we define a KT-invex control problem. We prove that a KT-invex control problem is characterized in order that a Kuhn–Tucker point is an optimal solution. We generalize optimality results of known mathematical programming problems. We illustrate these results with examples.

Multiobjective fractional programming with generalized convexity

This paper derives several results regarding the optimality conditions and duality properties for the class of multiobjective fractional programs under generalized convexity assumptions. These results are obtained by applying a parametric approach to reduce the problem to a more conventional form.

On invex fuzzy mappings and fuzzy variational-like inequalities

In this paper, we first show the need for introducing invex fuzzy mappings. After that, we show that the concept of invex fuzzy mapping previously given by Wu and Xu are very restrictive and the examples presented there are not correct. Then, we present more general concepts of invex and incave fuzzy mappings involving strongly generalized differentiable fuzzy mapping. Finally, we show that the results obtained by Wu and Xu on the relationship between fuzzy variational-like inequalities and fuzzy optimization problems are still valid using these new concepts.

s-Convex fuzzy processes

Abstract We introduce the notion of s -convex fuzzy processes. We study their properties and we give some applications.

Necessary and sufficient conditions for fuzzy optimality problems

In this paper we define a new minimum concept for fuzzy optimization problems more general than those that exist in the literature. We find necessary optimality conditions based on a new fuzzy stationary point definition. And we prove that these conditions are also sufficient under new fuzzy generalized convexity notions.

A Note on Generalized Convexity for Fuzzy Mappings Through a Linear Ordering

In De Campos Ibanez and Gonzalez-Munoz (Fuzzy Sets Syst 29:145–154, 1989, [6]), Goestschel and Voxman (Fuzzy Sets Syst 18:31–43, 1986, [7]) the authors considered a linear ordering on the space of fuzzy intervals. For each fuzzy mapping (fuzzy interval-valued mapping) F, based on the aforementioned linear ordering, they introduced a real-valued function \(T_F\) on the domain of the fuzzy mapping F. Recently, Chalco-Cano et al. (Fuzzy Sets Syst 231:70–83, 2013, [4]) have studied the relationship between the generalized Hukuhara differentiability of a fuzzy mapping F (G-differentiability, for short) and the differentiability of \(T_F\), and some properties of local-global minima. This paper studies such properties for fuzzy mappings, using new concepts which generalize the existing ones.

Weak efficiency in multiobjective variational problems under generalized convexity

In this paper, we provide new pseudoinvexity conditions on the involved functionals of a multiobjective variational problem, such that all vector Kuhn-Tucker or Fritz John points are weakly efficient solutions if and only if these conditions are fulfilled. We relate weakly efficient solutions to optimal solutions of weighting problems. We improve recent papers, and we generalize pseudoinvexity conditions used in multiobjective mathematical programming, so as some of their characterization results. The new conditions and results are illustrated with an example.