A Note on the Characterization of the Global Maxima of a (Tangentially) Convex Function Over a Convex Set

We consider the problem of characterizing points x̄ in a convex set C which globally maximize an objective function f over C. When f is convex, we show how the first order necessary condition extended to those x ∈ C at the same level as x̄ is necessary and sufficient for x̄ being a global maximum of f over C.This improves a recent result by A. Strekalovski who was the first to propose such type of global optimality condition, but whose characterization required to scan all the points (even out of C) at the same level as x̄. Next we extend the obtained characterization to the case where the objective function is just tangentially convex and an appropriate qualification condition holds.