Carathéodory's lemma states that if we have a linear combination of vectors in <img border=0 src="../../../../img/revistas/cam/v29n2/r_bastao.gif" align=absmiddle>n, we can rewrite this combination using a linearly independent subset. This lemma has been successfully applied in nonlinear optimization in many contexts. In this work we present a new version of this celebrated result, in which we obtained new bounds for the size of the coefficients in the linear combination and we provide examples where these bounds are useful. We show how these new bounds can be used to prove that the internal penalty method converges to KKT points, and we prove that the hypothesis to obtain this result cannot be weakened.The new bounds also provides us some new results of convergence for the quasi feasible interior point ℓ2-penalty method of Chen and Goldfarb [7]. Mathematical subject classification: 90C30, 49K99, 65K05.
nonlinear programming; constraint qualifications; interior point methods
