This paper focuses on formulas for the ε-subdifferential of the optimal value function of
scalar and vector convex optimization problems. These formulas can be applied when the
set of solutions of the problem is empty. In the scalar case, both unconstrained problems and
problems with an inclusion constraint are considered. For the last ones, limiting results are
derived, in such a way that no qualification conditions are required. The main mathematical
tool is a limiting calculus rule for the ε-subdifferential of the sum of convex and lower
semicontinuous functions defined on a (non necessarily reflexive) Banach space. In the
vector case, unconstrained problems are studied and exact formulas are derived by linear
scalarizations. These results are based on a concept of infimal set, the notion of cone proper
set and an ε-subdifferential for convex vector functions due to Taa.
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