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The growing use of optimization models to help decision making has created a demand for such tools that allow formulating and solving more models of real-world processes and systems related to human activity in which hypotheses are not verify in a way specific for classical optimization. One of the approaches for real-world extremum problems under uncertainty is interval-valued optimization. In this paper, a twice differentiable vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. In this paper, the first order necessary optimality conditions of Karush-Kuhn-Tucker type are proved for differentiable interval-valued vector optimization problems under the first order constraint qualification. If the interval-valued objective function is assumed to be twice weakly differentiable and constraints functions are assumed to be twice differentiable, then two types of second order necessary optimality conditions under two various constraint qualifications are proved for such smooth interval-valued vector optimization problems. Finally, in order to illustrate the Karush-Kuhn-Tucker type necessary optimality conditions established in the paper, an example of an interval-valued optimization is given. [ABSTRACT FROM AUTHOR]

In this paper, two mathematical methods are used for solving a complex multicriteria optimization problem as the considered convex differentiable vector optimization problem with vanishing constraints. First of them is the linearized approach in which, for the original vector optimization problem with vanishing constraints, its associated multiobjective programming problem is constructed at the given feasible solution. Since the aforesaid multiobjective programming problem constructed in the linearized method is linear, one of the existing methods for solving linear vector optimization problems is applied for solving it. Thus, the procedure for solving the considered differentiable vector optimization problems with vanishing constraints is presented in the paper. [ABSTRACT FROM AUTHOR]

In this article, we use the so-called η-approximation method for solving a new class of nonconvex variational problems with exponential (p, r)-invex functionals. In this approach, we construct η-approximated variational problem and η-approximated Mond- Weir dual variational problem for the considered variational problem and its Mond-Weir dual variational problem. Then several duality results for considered variational problem and its Mond-Weir dual variational problem are proved by the help of duality results established between η-approximated variational problems mentioned above.

Optimization problems under uncertainty has achieved much attention because of inaccurate or incomplete data that often appear in real-world applications. In order to address this challenging issue, several types of methodologies have been developed in optimization theory and one of them is fuzzy optimization. In this paper, an attempt is taken to use the minimax exact penalty function method for solving convex nondifferentiable optimization problems with fuzzy objective functions and with both inequality and equality constraints. The most important property of exact penalty function methods, that is, exactness of the penalization, is defined and analyzed if the minimax penalty function method is applicable for solving a convex nondifferentiable optimization problem with a fuzzy objective function. It is proved that a (weak) KKT point of the considered fuzzy optimization problem is a (weakly) nondominated solution of its associated fuzzy penalized optimization problem constructed in the used penalized approach. Further, the conditions are also derived under which there is the equivalence between (weakly) nondominated solutions of the aforesaid fuzzy optimization problems. These results are established under assumption that the functions involved in the considered nondifferentiable fuzzy optimization problem are convex. Further, the algorithm of solving the considered nondifferentiable fuzzy optimization problem by using the minimax exact penalty fuzzy function method is proposed and its convergence is also established. [ABSTRACT FROM AUTHOR]

In the paper, we investigate a class of nondifferentiable semi-infinite multiobjective programming problems such that all functions constituting them are interval-valued. We derive both Fritz John necessary optimality conditions and, under a constraint qualification, Karush-Kuhn-Tucker necessary optimality conditions for (weak) $ LU $-Pareto solutions in the considered nondifferentiable semi-infinite vector interval-valued optimization problem. Under appropriate invexity hypotheses, we also prove sufficient optimality conditions for this semi-infinite interval-valued vector optimization problem. Further, we also use the $ l_{1} $ exact function method for solving the aforesaid nondifferentiable interval-valued multicriteria optimization problem. Then, we analyze the property of exactness of the penalization for the absolute value exact penalty function method under assumption that the functions involved in the considered semi-infinite multiobjective programming problem are locally Lipschitz interval-valued invex functions. The conditions guaranteeing the equivalence of the sets of (weak) $ LU $-Pareto solutions in the original semi-infinite interval-valued multiobjective programming problem and its associated vector penalized optimization problem with the multiple interval-valued $ l_{1} $ exact penalty function are given. [ABSTRACT FROM AUTHOR]