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Counterexamples to a triality theorem for quadratic-exponential minimization problems

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Abstract

The main goal of this note is to give a counterexample to the Triality Theorem in Gao and Ruan (Math Methods Oper Res 67:479–491, 2008). This is done first by considering a more general optimization problem with the aim to encompass several examples from Gao and Ruan (Math Methods Oper Res 67:479–491, 2008) and other papers by Gao and his collaborators (see f.i. Gao Duality principles in nonconvex systems. Theory, methods and applications. Kluwer, Dordrecht, 2000; Gao and Sherali Advances in applied mathematics and global optimization. Springer, Berlin, 2009). We perform a thorough analysis of the general optimization problem in terms of local extrema while presenting several counterexamples.

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References

  • Gao DY (2000) Duality principles in nonconvex systems. Theory, methods and applications. Kluwer, Dordrecht

  • Gao DY, Ruan N (2008) Solutions and optimality criteria for nonconvex quadratic-exponential minimization problem. Math Methods Oper Res 67:479–491

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  • Gao DY, Sherali HD (2009) Canonical duality: connection between nonconvex mechanics and global optimization. In: Advances in applied mathematics and global optimization. Springer, pp 249–316

  • Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton

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  • Zălinescu C (2002) Convex analysis in general vector spaces. World Scientific, Singapore

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Acknowledgments

The authors thank one of the anonymous referees for his/her comments and suggestions which helped to improve the presentation of the article. Moreover, the authors wish to mention that an earlier version of the paper was submitted to the journal on September 11th, 2009.

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Authors and Affiliations

  1. Towson University, Towson, MD, USA

    M. D. Voisei

  2. Faculty of Mathematics, University Alexandru Ioan Cuza, Iasi, Romania

    C. Zălinescu

  3. Octav Mayer Institute of Mathematics (Romanian Academy), Iasi, Romania

    C. Zălinescu

Authors
  1. M. D. Voisei
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  2. C. Zălinescu
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Correspondence to C. Zălinescu.

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Voisei, M.D., Zălinescu, C. Counterexamples to a triality theorem for quadratic-exponential minimization problems. Math Meth Oper Res 77, 227–237 (2013). https://doi.org/10.1007/s00186-012-0426-1

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  • DOI: https://doi.org/10.1007/s00186-012-0426-1

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