Optimality Conditions in Vector Optimization

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We present new classes of vector invex and pseudoinvex functions which generalize the class of scalar invex functions. These new classes of vector functions are characterized in such a way that every vector critical point is an efficient or a weakly efficient solution of a Multiobjective Programming Problem. We establish relationships between these new classes of functions and others used in the study of efficient and weakly efficient solutions, by the introduction of several examples. These results and classes of vector functions are extended to the involved functions in constrained multiobjective mathematical programming problems. It is proved that in order for Kuhn-Tucker points to be efficient or weakly efficient solutions it is necessary and sufficient that the multiobjective problem functions belong to a new class of functions, which we introduce. Similarly, we present characterizations for efficient and weakly efficient solutions by using Fritz John optimality conditions. Some examples are proposed to illustrate these classes of functions and optimality results.

We propose a unifying approach in deriving constraint qualifications and theorem of the alternative. We first introduce a separation theorem between a subspace and the non-positive orthant, and then we use it to derive a new constraint qualification for a smooth vector optimization problem with inequality constraints. The proposed condition is weaker than the existing conditions stated in the recent literature. According with the strict relationship between generalized convexity and constraint qualifications, we introduce a new class of generalized convex vector functions. This allows us to obtain some new constraint qualifications in a more general form than the ones related to componentwise generalized convexity. Finally, the introduced separation theorem allows us to derive some of the known theorems of the alternative which are used in the literature to get constraint qualifications.

We are interested in proving optimality conditions for optimization problems. By means of different second-order tangent sets, various second-order necessary optimality conditions are obtained for both scalar and vector optimization problems where the feasible region is given as a set. We present also second-order sufficient optimality conditions so that there is only a very small gap with the necessary optimality conditions. As an application we establish second-order optimality conditions of Fritz John type, Kuhn-Tucker type 1, and Kuhn-Tucker type 2 for a problem with both inequality and equality constraints and a twice differentiable functions. At the end, a very general second-order necessary conditions for efficiency with respect to cones is present and it is applied to smooth and nonsmooth data.

In this work we study the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems. We consider problems whose objective functions are defined between infinite and finite-dimensional Banach spaces. Our results are stated under hypotheses of generalized convexity and make use of variational-like inequalities.

In this paper, the concept of proper eciency has been incorporated to develop the optimality conditions and duality results for dierentiable nonconvex multiobjective programming problems. The assumptions on properly eciency are relaxed by means of the introduced classes of vector-valued B-(p; r)-invex functions. These results ex- tend several known results to a wider class of nonconvex vector optimization problems.

The goal of this paper is to obtain Kuhn-Tucker type necessary and sucient op- timality conditions for nonsmooth constrained optimization problems involving gen- eralized type-I functions on the objective and constraint functions involved in the problem. Further we study the duality results in the presence of the aforesaid weaker assumptions.

In this chapter we present and prove dierent forms of weak, strong and converse duality theorems for the Wolfe and Mond-Weir dual problems associated with the vector optimization problem with constraints, where the vector objective function and the vector function associated with the inequality-type constraints are invex, strictly invex or quasi-invex.

The aim of this chapter is to propose some pairs of dual programs where the pri- mal is a vector problem having a feasible region defined by a set constraint, equality and inequality constraints, while the duals can be classified as \mixed type" ones. The duality results are proved under suitable generalized concavity properties. In this light, the role of dierent kinds of generalized concavity properties will be deepened on.

We discuss necessary and sucient conditions of optimality for nonsmooth and smooth continuous-time multiobjective optimization problems under generalized convexity assumptions.

In this chapter we present some classes of nonsmooth continuous-time problems. Optimality conditions under certain structure of generalized convexity are derived for these classes. Subsequently, two dual models are formulated and weak and strong duality theorems are established.

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