25 research outputs found
Decision analysis: vector optimization theory
First published in Revista de la Real Academia de Ciencias Exactas, F铆sicas y Naturales in 93, 4, 1999, published by the Real Academia de Ciencias Exactas, F铆sicas y Naturales
Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions
The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush-Kuhn-Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash's critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces
Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions
In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results
Generalized convexity: Their applications to variational problems
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
Optimality and duality on Riemannian manifolds
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.Ministerio de Econom铆a y Competitivida
Generalized convexity: Their applications to multiobjective programming
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
Semi-infinite interval equilibrium problems: optimality conditions and existence results
This paper aims to obtain new Karush鈥揔uhn鈥揟ucker optimality conditions for solutions
to semi-infinite interval equilibrium problems with interval-valued objective functions. The
Karush鈥揔uhn鈥揟ucker conditions for the semi-infinite interval programming problem are
particular cases of those found in this paper for constrained equilibrium problem.We illustrate
this with some examples. In addition, we obtain solutions to the interval equilibrium problem
in the unconstrained case. The results presented in this paper extend the corresponding results
in the literature.The research has been supported by FEDER Andaluc铆a 2014-2020 (UPO-1381297) and by MICIN through grant MCIN/AEI/PID2021-123051NB-100
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鈥檚 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.The research in this paper has been supported by MTM2015-66185 (MINECO/FEDER, UE) and
Fondecyt-Chile, Project 1151154
The continuous-time problem with interval-valued functions: applications to economic equilibrium
The aim of this paper is to define the Continuous-Time Problem
in an interval context and to obtain optimality conditions for this
problem. In addition, we will find relationships between solutions
of Interval Continuous-Time Problem (ICTP) and Interval Variationallike
Inequality Problems, both Stampacchia and Minty type. Pseudo
invex monotonicity condition ensures the existence of solutions
of the (ICTP) problem. These results generalize similar conclusions
obtained in Euclidean or Banach spaces inside classical mathematical
programming problems or Continuous-Time Problems. We will finish
generalizing the existence of Walrasarian equilibrium price model
and the Wardrop鈥檚 principle for traffic equilibrium problem to an
environment of interval-valued functions.The research in this paper has been partially supported by Ministerio de Econom铆a y Competitividad,
Spain, through grant MTM2015-66185-P and Proyectos I+D 2015 MTM2015-66185-P
(MINECO/FEDER) and Fondecyt, Chile, grant 1151154
Second-order optimality conditions for interval-valued functions
This work is included in the search of optimality conditions for solutions to the scalar
interval optimization problem, both constrained and unconstrained, by means of
second-order optimality conditions. As it is known, these conditions allow us to reject
some candidates to minima that arise from the first-order conditions. We will define
new concepts such as second-order gH-derivative for interval-valued functions,
2-critical points, and 2-KKT-critical points. We obtain and present new types of
interval-valued functions, such as 2-pseudoinvex, characterized by the property that
all their second-order stationary points are global minima. We extend the optimality
criteria to the semi-infinite programming problem and obtain duality theorems.
These results represent an improvement in the treatment of optimization problems
with interval-valued functions.Funding for open access publishing: Universidad de C谩diz/CBUA. The research has been supported by MCIN through
grant MCIN/AEI/PID2021-123051NB-I00