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 Cádiz/CBUA. The research has been supported by MCIN through grant MCIN/AEI/PID2021-123051NB-I00

Optimal control and nonlinear programming

In this thesis, we have two distinct but related subjects: optimal control and nonlinear programming. In the first part of this thesis, we prove that the value function, propagated from initial or terminal costs, and constraints, in the form of a differential equation, satisfy a subgradient form of the Hamilton-Jacobi equation in which the Hamiltonian is measurable with respect to time. In the second part of this thesis, we first construct a concrete example to demonstrate conjugate duality theory in vector optimization as developed by Tanino. We also define the normal cones corresponding to Tanino\u27s concept of the subgradient of a set valued mapping and derive some infimal convolution properties for convex set-valued mappings. Then we deduce necessary and sufficient conditions for maximizing an objective function with constraints subject to any convex, pointed and closed cone

A Better Approach for Solving a Fuzzy Multiobjective Programming Problem by Level Sets

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

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’s principle for traffic equilibrium problem to an environment of interval-valued functions.The research in this paper has been partially supported by Ministerio de Economía y Competitividad, Spain, through grant MTM2015-66185-P and Proyectos I+D 2015 MTM2015-66185-P (MINECO/FEDER) and Fondecyt, Chile, grant 1151154

On optimality and duality for multiobjective interval-valued programming problems with vanishing constraints

In this study, we explore the theoretical features of a multiobjective interval-valued programming problem with vanishing constraints. In view of this, we have defined a multiobjective interval-valued programming prob-lem with vanishing constraints in which the objective functions are consid-ered to be interval-valued functions, and we define an LU-efficient solution by employing partial ordering relations. Under the assumption of general-ized convexity, we investigate the optimality conditions for a (weakly) LU-efficient solution to a multiobjective interval-valued programming problem with vanishing constraints. Furthermore, we establish Wolfe and Mond–Weir duality results under appropriate convexity hypotheses. The study concludes with examples designed to validate our findings

Some Aspects Of Duality In Variational Problems And Optimal Control

This thesis is divided into six chapters. In the Ist chapter we present a brief survey of related work done in the area of multiobjective mathematical programming, optimal control and game theory. Chapter Two: In this chapter sufficient optimality criteria are derived for a control problem under generalized invexity. A Mond-Weir type dual to the control problem is proposed and various duality theorems are validated under generalized invexity assumptions on functionals appearing in the problems. It is pointed out that these results can be applied to the control problem with free boundary conditions and have linkage with results for nonlinear programming problems in the presence of inequality and equality constraints already established in the literature. Chapter Three: In this chapter a mixed type dual to the control problem in order to unify Wolfe and Mond-Weir type dual control problem is presented in various duality results are validated and the generalized invexity assumptions. It is pointed out that our results can be extended to the control problems with free boundary conditions. The duality results for nonlinear programming problems already existing in the literature are deduced as special cases of our results. Chapter Four: In this chapter two types of duals are considered for a class of variational problems involving higher order derivative. The duality results are derived without any use of optimality conditions. One set of results is based on Mond-Weir type dual that has the same objective functional as the primal problem but different constraints. The second set of results is based on a dual of an auxiliary primal with single objective function. Under various convexity and generalized convexity assumptions, duality relationships between primal and its various duals are established. Problems with natural boundary values are considered and the analogues of our results in nonlinear programming are also indicated. Chapter Five: In this chapter a certain constrained dynamic game is shown to be equivalent to a pair of symmetric dual variational problems which have more general formulation than those already existing in the literature. Various duality results are proved under convexity and generalized convexity assumptions on the appropriate functional. The dynamic game is also viewed as equivalent to a pair of dual variational problems without the condition of fixed points. It is also indicated that our equivalent formulation of a pair of symmetric dual variational problems as dynamic generalization of those already studied in the literature. Chapter Six: In this chapter a mixed type second-order dual to a variational problem is formulated as a unification of Wolfe and Mond-Weir type dual problems already treated in the literature and various duality results are validated under generalized second order invexity. Problems with natural boundary values are formulated and it also is pointed out that our duality results can be regarded as dynamic generalizations of those of (static) nonlinear programming. The subject matter of the present research thesis is fully published in the form of the following research papers written by the author: (1) Sufficiency and Duality In Control Problems with Generalized Invexity, Journal of Applied Analysis,Vol, 14 No. 1 (2008),pp.27-42. (2) Mixed Type Duality for Control Problems with Generalized Invexity, Journal of Applied Mathematics and Informatics,Vol. 26(2008), No.5-6 , pp. 819-837. (3) On Multiobjective Duality for Variational Problems, The Open Operational Research Journal,2012, 6, 1-8. (4) Constrained Dynamic Game and Symmetric Duality For Variational Problems, Journal of Mathematics and System Science 2(2012), 171-178. (5) Mixed Type Second – Order Dulaity For Variational Problems, Journal of Informatics and Mathematical Sciences , Vol5,No.1, pp.1-13,(2013)

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 Economía y Competitivida