In this paper we deal with the following generalized vector variational inequality problem: let X, Y and Z be topological vector spaces, K be a convex set in X, D be a nonempty set in Y , and C be a closed convex cone in Z. Let T : K → 2 be a multifunction and f : K ×D ×K → Z be a single-valued mapping. Find a point x̂ ∈ K and ŷ ∈ T (x̂) such that f(x̂, ŷ, z) / ∈ −intC, ∀z ∈ K. We prove some exist...

In this paper we introduce notions of well-posedness for a vector optimization problem and for a vector variational inequality of differential type, we study their basic properties and we establish the links among them. The proposed concept of well-posedness for a vector optimization problem generalizes the notion of well-setness for scalar optimization problems, introduced in [2]. On the other...

In this paper, we provide a couple of solutions for the vector variational inequality problem, adopting topological approach. We consider here more general framework, where X and Y are spaces. Topological concepts including continuity, compactness, closedness, so on used obtaining our results. The condition admissibility function space topology is found to play an important role in achieving It...

For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D-gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization-based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D-gap functions for a quasi-vari...

In this note we consider some notions of well-posedness for scalar and vector variational inequalities and we recall their connections with optimization problems. Subsequently, we investigate similar connections between well-posedness of a vector optimization problem and a related variational inequality problem and we present a result obtained with scalar characterizations of vector optimality ...

We consider variational inequality problems for set-valued vector fields on general Riemannian manifolds. The existence results of the solution, convexity of the solution set, and the convergence property of the proximal point algorithm for the variational inequality problems for set-valued mappings on Riemannian manifolds are established. Applications to convex optimization problems on Riemann...

Let E be a topological vector space and X be a non-empty subset of E. Let S : X → 2X and T : X → 2E be two maps. Then the generalized quasi-variational inequality (GQVI) problem is to find a point ŷ ∈ S(ŷ) and a point ŵ ∈ T (ŷ) such that Re〈ŵ, ŷ − x〉 ≤ 0 for all x ∈ S(ŷ). We shall use Chowdhury and Tan’s 1996 generalized version of Ky Fan’s minimax inequality as a tool to obtain some general th...

We present the technical background material for a version of the Pontryagin Maximum Principle with state space constraints and very weak technical hypotheses, based on a primal approach that uses generalized differentials and packets of needle variations. In particular, we give a detailed account of two theories of generalized differentials, the " generalized differential quotients " (GDQs) an...