On a Conjecture in Second-Order Optimality Conditions
In this paper, we deal with a conjecture formulated in Andreani et al. (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. We prove the conjecture under the additional assumption that the Jacobian matrix has a smooth singular value decomposition. Our proof also extends to the case of the strong second-order condition, defined in terms of the critical cone instead of the critical subspace.
KeywordsNonlinear optimization Constraint qualifications Second-order optimality conditions Singular value decomposition
Mathematics Subject Classification90C46 90C30 15B10
This research was partially conducted while the second author held a Visiting Scholar position at Department of Management Science and Engineering, Stanford University, Stanford CA, USA. The fourth author was a Ph.D. student at Department of Applied Mathematics, University of São Paulo-SP, Brazil. We thank the referees and the handling editor for insightful remarks. This research was funded by FAPESP Grants 2013/05475-7 and 2016/02092-8, by CNPq Grants 454798/2015-6, 303264/2015-2 and 481992/2013-8, and CAPES.
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