On a Conjecture in Second-Order Optimality Conditions

Roger Behling (), Gabriel Haeser (), Alberto Ramos () and Daiana S. Viana ()
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Roger Behling: Federal University of Santa Catarina
Gabriel Haeser: University of São Paulo
Alberto Ramos: Federal University of Paraná
Daiana S. Viana: Federal University of Acre

Journal of Optimization Theory and Applications, 2018, vol. 176, issue 3, No 6, 625-633

Abstract: Abstract 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.

Keywords: Nonlinear optimization; Constraint qualifications; Second-order optimality conditions; Singular value decomposition; 90C46; 90C30; 15B10 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10957-018-1229-1

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