  There are several concepts and definitions that characterize and give
optimality conditions for solutions of a vector optimization problem. One of
the most important is the first-order necessary optimality condition that
generalizes the Karush-Kuhn-Tucker condition. This condition ensures the
existence of an arbitrary neighborhood that contains an local optimal solution.
The present work we introduce an alternative concept to identify the local
optimal solution neighborhood in vector optimization problems. The main aspect
of this contribution is the development of necessary and sufficient Pareto
optimality conditions for the solutions of a particular vector optimization
problem, where each objective function consists of a ratio quadratic functions
and the feasible set is defined by linear inequalities. We show how to
calculate the largest radius of the spherical region centered on a local Pareto
solution in which this solution is optimal. In this process we may conclude
that the solution is also globally optimal. These conditions might be useful to
determine termination criteria in the development of algorithms, including more
general problems in which quadratic approximations are used locally.
