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In this paper, a new generalized Hukuhara differentiability concept for interval-valued functions defined on Rn is proposed, which extends the classical Fréchet differentiability notion and provides an interval quasilinear approximation for an interval-valued function in a neighborhood of a point at which such function is gH-differentiable. Moreover, it overcomes the shortcomings generated by the use of the gH-differentiability concept previously presented in the literature, and this presents a good perspective on interval and fuzzy environments. Several properties of this new concept are investigated and compared with the previous concept properties. Furthermore, the gH-differentiability concept is extended for a fuzzy function, and its introduction is argued and illustrated with examples.

In this paper we present a review of the most important notions and characterizations of differentiability and necessary optimality conditions for a fuzzy multiobjective problem. As basis of this review, we first study the fundamental aspects of the notions of differentiability for interval valued functions, since the fuzzy environment and the interval environment are closely related. Those aspects are related to the different definitions of difference for intervals and their drawbacks, the different definitions and characterizations of the differentiability for interval-valued functions and their drawbacks and how they have been solved in the literature. Based on the most important and meaning results on interval valued functions you can find in the literature, a review on notions of differentiability in fuzzy context is given, both in the case of functions of one variable, and several variables. And finally we present the review results of the necessary optimality conditions for fuzzy multiobjective problems and the main conclusions.

In this paper, we deal with the resolution of a fuzzy multiobjective programming problem using the level sets optimization. We compare it to other optimization strategies studied until now and we propose an algorithm to identify possible Pareto efficient optimal solutions.

This article has two objectives. Firstly, we use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of generalized approximate geodesic convex functions and illustrated them with examples. We see the minimum requirements under which critical points, solutions of Stampacchia, and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we show an economical application, again using solutions of the variational problems to identify Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

If x∗ is a local minimum solution, then there exists a ball of radius r > 0 such that f (x) ≥ f (x∗) for all x ∈ B(x∗,r). The purpose of the current study is to identify the suitable B(x∗,r) of the local optimal solution x∗ for a particular multiobjective optimization problem. We provide a way to calculate the largest radius of the ball centered at local Pareto solution in which this solution is optimal. In this process, we present the necessary and sufficient conditions for achieving a global Pareto optimal solution. The results of this investigation might be useful to determine stopping criteria in the algorithms development.

Our goal in this paper is to translate results on function classes that are characterized by the property that all the Karush-Kuhn-Tucker points are efficient solutions, obtained in Euclidean spaces to Riemannian manifolds. We give two new characterizations, one for the scalar case and another for the vectorial case, unknown in this subject literature. We also obtain duality results and give examples to illustrate it.

In this paper, we focus on necessary and sufficient efficiency conditions for optimization problems with multiple objectives and a feasible set defined by interval-valued functions. A new concept of Fritz-John and Karush–Kuhn–Tucker-type points is introduced for this mathematical programming problem based on the gH-derivative concept. The innovation and importance of these concepts are presented from a practical and computational point of view. The problem is approached directly, without transforming it into a real-valued programming problem, thereby attaining theoretical results that are more powerful and computationally more efficient under weaker hypotheses. We also provide necessary conditions for efficiency, which have been inexistent in the relevant literature to date. The identification of necessary conditions is important for the development of future computational optimization techniques in an interval-valued environment. We introduce new generalized convexity notions for gH-differentiable interval-valued problems which are a generalization of previous concepts and we prove a sufficient efficiency condition based on these concepts. Finally, the efficiency conditions for deterministic programming problems are shown to be particular instances of the results proved in this paper. The theoretical developments are illustrated and justified through several numerical examples.

The aim of this paper is to show some applicable results to multiobjective optimization problems and the role that the Generalized Convexity plays in them. The study of convexity for sets and functions has special relevance in the search of optimal functions, and in the development of algorithms for solving optimization problems. However, the absence of convexity implies a total loss of effectiveness of the Optimization Theory methods, ie, the results are being verified under less stringent conditions, it was what became known as Generalized convexity. The literature generated around this topic has demonstrated its importance both from a theoretical point of view as practical, but it has also generated an enormous amount of papers with little scientific input.

The aim of this paper is to show one of the generalized convexity applications, generalized monotonicity particularly, to the variational problems study. These problems are related to the search of equilibrium conditions in physical and economic environments. If convexity plays an important role in mathematical programming problems, monotonicity will play a similar role in variational problems. This paper shows some recent results about this topic.

This paper introduces a new condition on the functionals of a control problem and extends a recent characterization result of KT-invexity. We prove that the new condition, the FJinvexity, is both necessary and sufficient in order to characterize the optimal solution set using Fritz John points.