We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qu...

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...

We propose to solve generalized bilevel programs by a trust region approach where the “model” takes the form of a bilevel program involving a linear program at the upper level and a linear variational inequality at the lower level. By coupling the concepts of trust region and linesearch in a novel way, we obtain an implementable algorithm that converges to a strong stationary point of the origi...

We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities, which includes several classes of variational inequalities and variational-like inequalities as special cases. By applying the auxiliary principle technique and KKM theory, we suggest an iterative algorithm for solving the generalized strongly nonlinear mixed variationallike inequality. The exi...

The aim of this paper is to introduce the generalized viscosity implicit rules of one nonexpansive mapping in uniformly smooth Banach spaces. Strong convergence theorems of the rules are proved under certain assumptions imposed on the parameters. As applications, we use our main results to solve fixed point problems of strict pseudocontractions in Hilbert spaces and variational inequality probl...

In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ...