Increase-along-rays property for vector functions (original) (raw)
Related papers
Increasing Along Rays Vector Functions
The class of increasing along rays functions is generalized to con-sider vector valued functions. A general approach through scalarization is used and minimal properties for the scalarization are given. The class of vector in-creasing along rays functions introduced is compared with the scalar one to prove similar properties hold. The relation with convex and generalized convex functions is preserved for the vector valued counterpart.
A note on Minty type vector variational inequalities
RAIRO - Operations Research, 2005
The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space Y are introduced. Under quasiconvexity assumptions we show that solutions of generalized variational inequality and of the primitive optimization problem are equivalent. Finally, we discuss the possibility to generalize the scheme of this paper to get further applications in vector optimization.
Vector optimization and variational-like inequalities
2009
In this paper, some properties of pseudoinvex functions are obtained. We study the equivalence between different solutions of the vector variational-like inequality problem. Some relations between vector variational-like inequalities and vector optimization problems for non-differentiable functions under generalized monotonicity are established.
Scalarization Techniques for Set-Valued Vector Variational Type Inequalities
In this paper, we introduce two kinds of Stampacchia-type set valued vector variational-type inequalities by using four types of Stampacchia-type scalar variational inequalities and obtained the relations of the solution sets between six variational inequalities, which are more generalized results than those considered in . K ℝ spaces and showed relationship between variational like inequality problems and convex programming as well as with complementarity problems.
Set Optimization Meets Variational Inequalities
Springer Proceedings in Mathematics & Statistics, 2015
We study necessary and sufficient conditions to attain solutions of set-optimization problems in terms of variational inequalities of Stampacchia and Minty type. The notion of solution we deal with has been introduced in [18]. To define the set-valued variational inequality, we introduce a set-valued directional derivative and we relate it to the Dini derivatives of a family of scalar problems. The optimality conditions are given by Stampacchia and Minty type Variational inequalities, defined both by set-valued directional derivatives and by Dini derivatives of the scalarizations. The main results allow to obtain known variational characterizations for vector-valued optimization problems as special cases.
Filomat
In this paper by using the scalariation method we introduced the concept of relaxed K-preinvex set-valued maps and obtain some equivalence results of them in terms of normal subdifferential. Also, we consider generalized Minty variational-like inequalities and show that the set of solutions is equal to scalarized set-valued optimization problems?s solutions under generalized relaxed convexity assumptions.
Vector nonsmooth variational-like inequalities and optimization problems
Nonlinear Analysis: Theory, Methods & Applications, 2009
In this paper, we first extend the concepts of Clarke's generalized derivation and subdifferential for a function whose domain is a subset of a metrizable topological vector space. We also introduce and consider a new class of variational-like inequalities, which are called the vector nonsmooth variational-like inequalities. We also establish the relationship between the nonsmooth variational-like inequalities and vector optimization problems under some suitable conditions. In particular, our results extend and generalize the results of Mishra and Wang [