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Canonical duality for solving general nonconvex constrained problems

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Abstract

This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed.

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Notes

  1. Similarly, the canonical dual \(P^d({\varvec{\sigma }}, {{\varvec{\varsigma }}})\) in [12] is concave in \({\varvec{\sigma }}\) and \({{\varvec{\varsigma }}}\), respectively, but not concave in \(({\varvec{\sigma }}, {{\varvec{\varsigma }}}) \in \mathcal{S}^+_c\).

  2. The objectivity is only sufficient but not necessary for the canonical duality theory.

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Acknowledgments

This research was supported by US Air Force Office of Scientific Research under the grant AFOSR FA9550-10-1-0487. Comments and suggestions from two anonymous referees are sincerely acknowledged.

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Authors and Affiliations

  1. Department of Computer Control and Management Engineering, University of Rome Sapienza, via Ariosto 25, Rome, Italy

    Vittorio Latorre

  2. School of Science Information Technology and Engineering, Federation University Australia, Mt Helen, Victoria, 3350, Australia

    David Yang Gao

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  1. Vittorio Latorre
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  2. David Yang Gao
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Correspondence to David Yang Gao.

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Latorre, V., Gao, D.Y. Canonical duality for solving general nonconvex constrained problems. Optim Lett 10, 1763–1779 (2016). https://doi.org/10.1007/s11590-015-0860-0

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