On quadratically constrained quadratic optimization problems and canonical duality theory
In this work we provide a rigorous study of quadratically constrained quadratic optimization problems using the method suggested by the canonical duality theory (CDT). Then we compare our results with those obtained using CDT. We also point out unnecessarily strong assumptions (e.g. Ruan and Gao, in: Gao, Latorre, Ruan (eds) Canonical duality theory. Advances in mechanics and mathematics. Springer, Cham, vol 37, pp 315–338, 2017), non-standard definitions (e.g. Ruan and Gao, in: Gao, Latorre, Ruan (eds) Canonical duality theory. Advances in mechanics and mathematics. Springer, Cham, vol 37, pp 187–201, 2017), unclear statements (e.g. Gao and Ruan in J Glob Optim 47:463–484, 2010), false results (e.g. Gao et al. in J Glob Optim 45:473–497, 2009; Gao and Ruan 2010).
KeywordsQuadratic optimization problem Dual function Canonical duality theory Counterexample
We thank Prof. Marius Durea for reading a previous version of the paper and for his useful remarks, as well as Prof. Oleg A. Prokopyev, Editor in chief of Optimization Letters, and the two anonymous referees for their useful remarks and suggestions. The results of this paper were announced at the 4th International Conference on Nonlinear Analysis and Optimization (NAOP 2018), June 18–20, 2018, Zanjan, Iran. The work was funded by CNCS-UEFISCDI (Romania) under Grant Number PN-III-P4-ID-PCE-2016-0188.
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