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On a Conjecture in Second-Order Optimality Conditions

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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.

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Acknowledgements

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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Authors and Affiliations

  1. Federal University of Santa Catarina, Blumenau, SC, Brazil

    Roger Behling

  2. Department of Applied Mathematics, University of São Paulo, São Paulo, SP, Brazil

    Gabriel Haeser

  3. Department of Mathematics, Federal University of Paraná, Curitiba, PR, Brazil

    Alberto Ramos

  4. Center of Exact and Technological Sciences, Federal University of Acre, Rio Branco, AC, Brazil

    Daiana S. Viana

Authors
  1. Roger Behling
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  2. Gabriel Haeser
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  3. Alberto Ramos
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  4. Daiana S. Viana
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Correspondence to Gabriel Haeser.

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Behling, R., Haeser, G., Ramos, A. et al. On a Conjecture in Second-Order Optimality Conditions. J Optim Theory Appl 176, 625–633 (2018). https://doi.org/10.1007/s10957-018-1229-1

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  • DOI: https://doi.org/10.1007/s10957-018-1229-1

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