Differentiable Vector Optimization Problems Research Articles

A linearized approach for solving differentiable vector optimization problems with vanishing constraints

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.

On vector variational E-inequalities and differentiable vector optimization problem

AbstractIn this paper, in order to characterize optimality of a class of nonconvex differentiable multiobjective programming problems, we introduce two new types of vector variational-like inequalities, namely, a weak variational E-inequality and a vector variational E-inequality. Namely, under (strictly) E-convexity, we prove relationships between the aforesaid vector variational-like inequalities and differentiable vector optimization problems. Further, under (strictly) pseudo-E-convexity, we are in position to identify vector critical E-points, weak E-Pareto (E-Pareto) solutions of differentiable vector optimization problems and solutions of these introduced vector variational-like inequalities.

On first and second order multiobjective programming with interval-valued objective functions

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.

Optimality conditions and duality for E-differentiable multiobjective programming involving V-E-type I functions

In this paper, we introduce a new concept of sets and a new class of functions called alpha-E-invex sets and V-E-preinvex functions. Furthermore, a new concept of generalized convexity is introduced for (not necessarily) differentiable vector optimization problems. Namely, the concept of V-E-type I functions is defined for E-differentiable vector optimization problem. A number of sufficiency results are established under various types of (generalized) V-E-type I requirements. Moreover, several E-duality theorems in the sense of Mond–Weir are proved under appropriate (generalized) V-E-type I functions.

Optimality conditions in a class of generalized convex optimization problems with the multiple interval-valued objective function

In this paper, a new class of generalized convex optimization problems with a multiple interval-valued objective function is considered. The concepts of B-preinvexity, B-invexity, and the generalized of B-invexity are extended to interval-valued functions. Additionally, several properties of interval-valued B-preinvex and B-invex functions are investigated. The Karush–Kuhn–Tucker necessary optimality conditions for a (weak LU-Pareto) LU-Pareto solution are established for a differentiable vector optimization problem with a multiple interval-valued objective function. Furthermore, the sufficient optimality conditions for both LU and UC order relations are derived for such interval-valued vector optimization problems under appropriate (generalized) B-invexity hypotheses. Suitable examples are provided to illustrate the aforementioned results.

KT-E-invexity in E-differentiable vector optimization problems

In this paper, a new concept of generalized convexity is introduced for E-differentiable vector optimization problems. Namely, the concept of KT-E-invexity is defined for (not necessarily) differentiable vector optimization problems in which the functions involved are E-differentiable. The sufficiency of the so-called E-Karush-Kuhn-Tucker optimality conditions is established for the considered E-differentiable multiobjective programming problem under assumption that is KT-E-invex at an E-Karush-Kuhn-Tucker point. Further, the examples of KT-E-invex optimization problems with E-differentiable functions are constructed to illustrate the aforesaid results. Moreover, the so-called vector Mond-Weir E-dual problem is also derived for the considered E-differentiable vector optimization problem and several E-duality theorems in the sense of Mond-Weir are derived under KT-E-invexity hypotheses.2020 Mathematics Subject Classification: 90C26, 90C29, 90C30, 90C46.

A new approximation approach to optimality and duality for a class of nonconvex differentiable vector optimization problems

In this paper, a new approximation method for a characterization of (weak) Pareto solutions in some class of nonconvex differentiable multiobjective programming problems is introduced. In this method, an auxiliary approximated vector optimization problem is constructed at a given feasible solution of the original multiobjective programming problem. The equivalence between (weak) Pareto solutions of these two vector optimization problems is established under $$(\Phi ,\rho )$$ -invexity hypotheses. By using the introduced approximation method, it is shown in some cases that a nonlinear differentiable multiobjective programming problem can be solved by the help of some methods for solving a linear vector optimization problem. Further, the introduced approximation method is used in proving several duality results in the sense of Mond-Weir for the considered vector optimization problem.

Optimality and duality in nonsmooth vector optimization with non-convex feasible set

For a convex programming problem, the Karush–Kuhn–Tucker (KKT) conditions are necessary and sufficient for optimality under suitable constraint qualification. Recently, Suneja et al. [Am. J. Oper. Res. 6 (2013) 536–541] proved KKT optimality conditions for a differentiable vector optimization problem over cones in which they replaced the cone-convexity of constraint function by convexity of feasible set and assumed the objective function to be cone-pseudoconvex. In this paper, we have considered a nonsmooth vector optimization problem over cones and proved KKT type sufficient optimality conditions by replacing convexity of feasible set with the weaker condition considered by Ho [Optim. Lett. 11 (2017) 41–46] and assuming the objective function to be generalized nonsmooth cone-pseudoconvex. Also, a Mond–Weir type dual is formulated and various duality results are established in the modified setting.

The $ F $-objective function method for differentiable interval-valued vector optimization problems

<p style='text-indent:20px;'>In this paper, a differentiable vector optimization problem with the multiple interval-valued objective function and with both inequality and equality constraints is considered. The Karush-Kuhn-Tucker necessary optimality conditions are established for such a differentiable interval-valued multiobjective programming problem. Further, a new approach, called <inline-formula><tex-math id="M2">\begin{document}$ F $\end{document}</tex-math></inline-formula>-objective function method, is introduced for solving the considered differentiable vector optimization problem with the multiple interval-valued objective function. In this method, its associated vector optimization problem with the multiple interval-valued <inline-formula><tex-math id="M3">\begin{document}$ F $\end{document}</tex-math></inline-formula>-objective function is constructed. Their equivalence is established under <inline-formula><tex-math id="M4">\begin{document}$ F $\end{document}</tex-math></inline-formula>-convexity assumptions. It is shown that the introduced approach can be used to solve a nonlinear nonconvex interval-valued optimization problem. By using the introduced approximation method, it is also presented in some cases that a nonlinear nonconvex interval-valued optimization problem can be solved by the help of methods for solving linear interval-valued optimization problems.

Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function

In this paper, a nonconvex vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. The functions constituting it are not necessarily differentiable, but they are $ E $-differentiable. The so-called $ E $-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered $ E $-differentiable vector optimization problem with the multiple interval-valued objective function. Also the sufficient optimality conditions are derived for such interval-valued vector optimization problems under appropriate (generalized) $ E $-convexity hypotheses.

Second order modified objective function method for twice differentiable vector optimization problems over cone constraints

In the paper, a vector optimization problem with twice differentiable functions and cone constraints is considered. The second order modified objective function method is used for solving such a multiobjective programming problem. In this method, for the considered twice differentiable multi-criteria optimization problem, its associated second order vector optimization problem with the modified objective function is constructed at the given arbitrary feasible solution. Then, the equivalence between the sets of (weakly) efficient solutions in the original twice differentiable vector optimization problem with cone constraints and its associated modified vector optimization problem is established. Further, the relationship between an (weakly) efficient solution in the original vector optimization problem and a saddle-point of the second order Lagrange function defined for the modified vector optimization problem is also analyzed.

E-invexity and generalized E-invexity in E-differentiable multiobjective programming

In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems. For an E-differentiable function, the concept of E-invexity is introduced as a generalization of the E-differentiable E-convexity notion. In addition, some properties of E-differentiable E-invex functions are investigated. Furthermore, the so-called E-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered E-differentiable vector optimization problems with both inequality and equality constraints. Also, the sufficiency of the E-Karush-Kuhn-Tucker necessary optimality conditions are proved for such E-differentiable vector optimization problems in which the involved functions are E-invex and/or generalized E-invex.

On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems

ABSTRACTIn this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush–Kuhn–Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP.

A Note on the Guignard Constraint Qualification and the Guignard Regularity Condition in Vector Optimization

Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.

Characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method

In this paper, a new approximation method is introduced to characterize a so-called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with ( F , ρ ) -convex functions of order 2. In this method, an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point. In order to prove the equivalence between the original multiobjective programming problem and its associated F -approximated vector optimization problem, the suitable ( F , ρ ) -convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem.

A new characterization of (weak) Pareto optimality for differentiable vector optimization problems with [formula omitted]-invex functions

In this paper, the so-called η -approximation approach is used to characterize solvability (in Pareto sense) of a nonlinear multiobjective programming problem with G -invex functions with respect to the same function η . In this method, an equivalent η -approximated vector optimization problem is constructed by a modification of both the objective and the constraint functions in the original multiobjective programming problem at the given feasible point. Moreover, in order to find a (weak) Pareto optimal solution in the original multiobjective problem, it is sufficient to solve its associated η -approximated vector optimization problem equivalent to the original multiobjective programming problem in the sense discussed in this paper.

Regularity Conditions in Differentiable Vector Optimization Revisited

This work is concerned with differentiable constrained vector optimization problems. It focus on the intrinsic connection between positive linearly dependent gradient sets and the distinct notions of regularity that come to play in this context. The main aspect of this contribution is the development of regularity conditions, based on the positive linear dependence or independence of gradient sets, for problems with general nonlinear constraints, without any convexity hypothesis. Being easy to verify, these conditions might be useful to define termination criteria in the development of algorithms.

On G-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond---Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.

AN η-APPROXIMATION METHOD FOR NONSMOOTH MULTIOBJECTIVE PROGRAMMING PROBLEMS

In this paper, a new approach to a characterization of solvability of a nonlinear nonsmooth multiobjective programming problem with inequality constraints is introduced. A family of η -approximated vector optimization problems is constructed by a modification of the objective and the constraint functions in the original nonsmooth multiobjective programming problem. The connection between (weak) efficient points in the original nonsmooth multiobjective programming problem and its equivalent η -approximated vector optimization problems is established under V -invexity. It turns out that, in most cases, solvability of a nonlinear nonsmooth multiobjective programming problem can be characterized by solvability of differentiable vector optimization problems.

ON SECOND ORDER NECESSARY OPTIMALITY CONDITIONS FOR VECTOR OPTIMIZATION PROBLEMS

Second order necessary optimality condition for properly efficient solutions of a twice differentiable vector optimization problem is given. We obtain a nonsmooth version of the second order necessary optimality condition for properly efficient solutions of a nondifferentiable vector optimization problem. Furthermore, we prove a second order necessary optimality condition for weakly efficient solutions of a nondifferentiable vector optimization problem.