On the global convergence of interior-pointnonlinear programming algorithms

Abstract

Abstract. Carathéodory's lemma states that if we have a linear combination of vectors inR 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 2 -penalty method of Chen and Goldfarb [7].Mathematical subject classification: 90C30, 49K99, 65K05.

Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

“…See also [19]. If one considers equality constraints in (NSDP) separately, one should employ an adapted version of Carathéodory's Lemma that fixes a particular subset of vectors, which can be found in [8,Lem.…”

“…Then, due to the linearity of G, it is immediate to see that Robinson's CQ does not hold at x = 0. On the other hand, for any x ∈ R 2 and any orthogonal matrix E ∈ R 2×2 , note that regardless of the form of E as in (19), we have v11(x, E) = 0, v22(x, E) = 0, and…”

Two New Weak Constraint Qualifications and Applications

“…In this case, it is natural to assume that the optimization problem satisfies a sufficient interior property, that is, that every local minimizer can be arbitrarily approximated by strictly feasible points. It is known from [16] that CPLD together with such sufficient interior property is equivalent to MFCQ. Hence, it is fruitless to use CPLD to generalize results based on MFCQ in the context of interior point methods.…”