A New Bound on the Multipliers Given by Carathéodory´s Theorem and a Result on the Internal Penalty Method

Carathéodory's theorem for cones states that if we have a linear combination of vectors in Rn, we can rewrite this combination using a linearly independent subset. This theorem has been successfully applied in nonlinear optimization in many contexts. In this work we present a new version of this celebrated theorem, in which we prove a bound for the size of the scalars in the linear combination and we provide examples where this bound is useful. We also prove that the convergence property of the internal penalty method cannot be improved.