B. Hernández-Jiménez

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.

FJ-Invex control problem

Abstract 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 FJ-invexity, is both necessary and sufficient in order to characterize the optimal solution set using Fritz John points.

Duality and a Characterization of Pseudoinvexity for Pareto and Weak Pareto Solutions in Nondifferentiable Multiobjective Programming

In this paper, we unify recent optimality results under directional derivatives by the introduction of new pseudoinvex classes of functions, in relation to the study of Pareto and weak Pareto solutions for nondifferentiable multiobjective programming problems. We prove that in order for feasible solutions satisfying Fritz John conditions to be Pareto or weak Pareto solutions, it is necessary and sufficient that the nondifferentiable multiobjective problem functions belong to these classes of functions, which is illustrated by an example. We also study the dual problem and establish weak, strong, and converse duality results.

New efficiency conditions for multiobjective interval-valued programming problems

Abstract 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.

Optimality conditions for generalized differentiable interval-valued functions

In this paper we study the optimal solutions set for a generalized differentiable interval-valued function. Necessary and sufficient optimality conditions are established for gH-differentiable functions. Convexity assumptions that are necessary or required to ensure the characterization of the optimal solutions are weaker or less strict than those presented in previous works. These convexity assumptions are the weakest to characterize the optimal solutions set. Known results for classical non interval-valued optimization are particular cases of the ones proved here.

Characterization of optimal solutions for nonlinear programming problems with conic constraints

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Cravens notion of K-invexity function (when K is a cone in ℝ n ) and Martins notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.

Generalized convexity for non-regular optimization problems with conic constraints

In non-regular problems the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with conic constraints by Izmailov and Solodov (SIAM J Control Optim 40(4):1280–1295, 2001). They are based on the so-called 2-regularity condition of the constraints at a feasible point. It is well known that generalized convexity notions play a very important role in optimization for establishing optimality conditions. In this paper we give the concept of Karush–Kuhn–Tucker point to rewrite the necessary optimality condition given in Izmailov and Solodov (SIAM J Control Optim 40(4):1280–1295, 2001) and the appropriate generalized convexity notions to show that the optimality condition is both necessary and sufficient to characterize optimal solutions set for non-regular problems with conic constraints. The results that exist in the literature up to now, even for the regular case, are particular instances of the ones presented here.

Optimality Conditions for Non-Regular Optimal Control Problems and Duality

ABSTRACT We define a new class of optimal control problems and show that this class is the largest one of control problems where every admissible process that satisfies the Extended Pontryaguin Maximum Principle is an optimal solution of nonregular optimal control problems. In this class of problems the local and global minimum coincide. A dual problem is also proposed, which may be seen as a generalization of the Mond–Weir-type dual problem, and it is shown that the 2-invexity notion is a necessary and sufficient condition to establish weak, strong, and converse duality results between a nonregular optimal control problem and its dual problem. We also present an example to illustrate our results.

Optimality conditions for fuzzy constrained programming problems

Abstract This paper solves optimization problems where both the objective and constraints are given by fuzzy functions. In order to get it, we first prove that these problems are equivalent to optimization problems where the constraints functions are non-fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far. This new stationary point concept is based on the gH-differentiability and has many computational advantages that we describe. It is well-known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity. And we are in that case due to the fact that the fuzzy numbers (intervals) space is a nonlinear one. In this direction, the gH-derivative for fuzzy functions is a concept that is more general than Hukuhara and level-wise derivatives that are usually used in fuzzy optimization so far, in the sense that they can be applied to a wider number of fuzzy function classes than above concepts. With this new differentiability concept, we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far. Moreover, due to the fact that we do not have a linear space for fuzzy numbers, the convex concepts and generalized convex fuzzy function notion are very restrictive, also. This implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful.

Different optimum notions for fuzzy functions and optimality conditions associated

Fuzzy numbers have been applied on decision and optimization problems in uncertain or imprecise environments. In these problems, the necessity to define optimal notions for decision-maker’s preferences as well as to prove necessary and sufficient optimality conditions for these optima are essential steps in the resolution process of the problem. The theoretical developments are illustrated and motivated with several numerical examples.