On a conjecture in second-order optimality conditions
Abstract
In this paper we deal with optimality conditions that can be verified by a nonlinear optimization algorithm, where only a single Lagrange multiplier is avaliable. In particular, we deal with a conjecture formulated in [R. Andreani, J.M. Martinez, M.L. Schuverdt, "On second-order optimality conditions for nonlinear programming", Optimization, 56:529--542, 2007], which states that whenever a local minimizer of a nonlinear optimization problem fulfills the Mangasarian-Fromovitz Constraint Qualification and the rank of the set of gradients of active constraints increases at most by one in a neighborhood of the minimizer, a second-order optimality condition that depends on one single Lagrange multiplier is satisfied. This conjecture generalizes previous results under a constant rank assumption or under a rank deficiency of at most one. In this paper we prove the conjecture under the additional assumption that the Jacobian matrix has a smooth singular value decomposition, which is weaker than previously considered assumptions. We also review previous literature related to the conjecture.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2017
- DOI:
- 10.48550/arXiv.1706.07833
- arXiv:
- arXiv:1706.07833
- Bibcode:
- 2017arXiv170607833B
- Keywords:
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- Mathematics - Optimization and Control;
- 90C46;
- 90C30
- E-Print:
- Extended Technical Report