  In the present paper, the Polyak's principle, concerning convexity of the
images of small balls through C1,1 mappings, is employed in the study of vector
optimization problems. This leads to extend to such a context achievements of
local programming, an approach to nonlinear optimization, due to B.T. Polyak,
which consists in exploiting the benefits of the convex local behaviour of
certain nonconvex problems. In doing so, solution existence and optimality
conditions are established for localizations of vector optimization problems,
whose data satisfy proper assumptions. Such results are subsequently applied in
the analysis of welfare economics, in the case of an exchange economy model
with infinite-dimensional commodity space. In such a setting, the localization
of an economy yields existence of Pareto optimal allocations, which, under
certain additional assumptions, lead to competitive equilibria.
