On a conjecture in second-order optimality conditions

A Simple Canonical Form for Nonlinear Programming Problems and Its Use

“…Along the proof we resort to two linear algebraic lemmas, which are proved in the appendix. These lemmas are variations of results already presented in other references [2,5], and we provide their proofs to make the article self contained. The first step to prove Theorem 1 is to use Theorem 2 to reduce problem (1) to the canonical form (21).…”

“…For instance, the authors of [2] attempted to obtain an appropriate coordinate system, by using a version of the Singular Value Decomposition, and succeeded in proving new particular cases of Andreani's conjecture with this approach. However, they did not prove the conjecture because their decomposition is not as effective as the canonical form: it has "high order terms" in places in which the canonical decomposition has exact zeros, and the technicalities required to handle these terms precluded them from obtaining a proof for which they had found all the other ingredients.…”

“…The first step to prove Theorem 1 is to use Theorem 2 to reduce problem (1) to the canonical form (21). In order to that we note that the Mangasarian-Fromovitz constraint qualification requires that Dh(0) has rank m and recall that we use m + r to denote the rank of the matrix Dhg(0) in Equation (2). Therefore, by changing the order of the inequality constraints if necessary, we can assume that the matrix in Equation ( 23) in the statement of Theorem 2 has rank m + r. We can then use this theorem to analyze Andreani's conjecture and assume without loss of generality that our linear programming problem has the form (21), because in Section 2 we have shown that Equation ( 3) and the Mangasarian-Fromovitz constraint qualification are invariant under changes of coordinates.…”

“…The Mangasarian-Fromovitz constraint qualification implies that for every d there exists λ and µ > 0 such that S(d) ≥ 0 in Equation (3) (see [2].) Since this equation is equivalent to Equation (25), for every d there exists γ such that S d ≥ 0 in Equation (25), and in order to complete the proof we use the following lemma:…”

“…Unfortunately, even under conditions like Mangasarian-Fromovitz, the analysis of problem (1) can be tricky, and it is not uncommon to find articles in which incorrect results or proofs are presented, as pointed out in [2]. In fact, we are all human and our capability to deal with the details involved in the analysis of problem (1) is limited, and we need tools to handle them.…”