355 research outputs found
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
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.The research in this paper has been supported by MTM2015-66185 (MINECO/FEDER, UE) and
Fondecyt-Chile, Project 1151154
Optimality Conditions For Multi-Objective Interval-Valued Optimization Problem On Hadamard Manifolds
The KKT optimality conditions for multi-objective interval-valued
optimization problem on Hadamard manifold are studied in this paper. Several
concepts of Pareto optimal solutions, considered under LU and CW ordering on
the class of all closed intervals in , are given. The KKT
conditions are presented under the notions of convexity, pseudo-convexity and
generalized Hukuhara difference. We show, with the help of an example, that the
results done in this paper for solving multi-objective interval-valued
optimization problems on Hadamard spaces are more general than the existing
ones on Euclidean spaces. The main results are supported by examples
Fuzzy Bilevel Optimization
In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems.
In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1
1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2
1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Preliminaries 11
2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11
2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
3 Optimization problem with fuzzy objective 19
3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25
4 Linear optimization with fuzzy objective 27
4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36
4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
5 Optimality conditions 47
5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48
5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49
5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51
5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52
5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54
5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Fuzzy linear optimization problem over fuzzy polytope 59
6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65
6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Bilevel optimization with fuzzy objectives 73
7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78
7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87
8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9 Conclusions 95
Bibliography 9
On the Newton method for solving fuzzy optimization problems
In this article we consider optimization problems where the objectives are fuzzy functions (fuzzy-valued functions). For this class of fuzzy optimization problems we discuss the Newton method to find a non-dominated solution. For this purpose, we use the generalized Hukuhara differentiability notion, which is the most general concept of existing differentiability for fuzzy functions. This work improves and correct the Newton Method
previously proposed in the literature.Fondo Nacional de Desarrollo Científico y Tecnológico (Chile)Ministerio de Ciencia y TecnologíaConselho Nacional de Desenvolvimento Científico e Tecnológico (Brasil)Centro de Pesquisa em Matemática Aplicada à Indústria (Fundação de Amparo à Pesquisa do Estado de São Paulo
Modern Approach for Designing and Solving Interval Estimated Linear Fractional Programming Models
Optimization methods have been widely applied in statistics. In mathematical programming, the coefficients of the models are always categorized as deterministic values. However uncertainty always exists in realistic problems. Therefore, interval-estimated optimization models may provide an alternative choice for considering the uncertainty into the optimization models. In this aspect, this paper concentrates, the lower and upper values of interval estimated linear fractional programming model (IELFPM) are obtained by using generalized confidence interval estimation method. An IELFPM is a LFP with interval form of the coefficients in the objective function and all requirements. The solution of the IELFPM is also analyzed
Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds
This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivatives, second-order pseudoinvexity functions, and the second-order Karush-Kuhn-Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization, respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of "Higgs Boson like" potentials, among others
- …