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An Extension of Yuan’s Lemma and Its Applications in Optimization

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Abstract

We prove an extension of Yuan’s lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of Baccari and Trad (SIAM J Optim 15(2):394–408, 2005), where the classical necessary second-order optimality condition is proved, under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the Hessian of the Lagrangian, evaluated at the vertices of the Lagrange multiplier set, is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the Jacobian matrix.

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Acknowledgements

This work was conducted with financial support by FAPESP (Grants 2013/05475-7 and 2016/02092-8) and CNPq, while the author was holding a Visiting Scholar position at Department of Management Science and Engineering, Stanford University, Stanford, CA, USA. The author would like to acknowledge the valuable comments and suggestions made by the referees.

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

    Gabriel Haeser

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  1. Gabriel Haeser
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Correspondence to Gabriel Haeser.

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Communicated by Juan-Enrique Martínez-Legaz.

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Haeser, G. An Extension of Yuan’s Lemma and Its Applications in Optimization. J Optim Theory Appl 174, 641–649 (2017). https://doi.org/10.1007/s10957-017-1123-2

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  • DOI: https://doi.org/10.1007/s10957-017-1123-2

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