  When optimization theorists consider optimization problems in infinite
dimensional spaces, they need to deal with closed convex subsets(usually cones)
which mostly have empty interior. These subsets often prevent optimization
theorists from applying powerful techniques to study these optimization
problems. In this paper, by nonsupport point, we present generating spaces
which are relative to a Banach space and a nonsupport point of its convex
closed subset. Then for optimization problems in infinite dimensional spaces,
in some general cases, we replace original spaces by generating spaces while
containing solutions. Thus this method enable us to apply powerful classical
techniques to optimization problems in very general class of infinite
dimensional spaces. Based on functional analysis, from classical Banach spaces
to separable Banach spaces, from Banach lattice to latticization, we give
characterizations of generating spaces and conclude that they are actually
linearly isometric to $L_\infty$($\ell _\infty$) or their closed subspaces.
Thus continuous linear functional involved in these techniques could be chosen
from $L_\infty^*$($\ell_\infty^*$). After that, applications in Penalty
principle, Lagrange duality and scalarization function are further studied by
this method.

Functional Analysis;Optimization and Control