On quadratically constrained quadratic optimization problems and canonical duality theory
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On quadratically constrained quadratic optimization problems and canonical duality theory 1,2 Constantin Zalinescu ˘
Received: 17 April 2019 / Accepted: 5 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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). Keywords Quadratic optimization problem · Dual function · Canonical duality theory · Counterexample
1 Introduction In the preface of the recent book [9], Gao’s canonical duality theory (CDT) is presented as being “a breakthrough methodological theory that can be used not only for modeling complex systems within a unified framework, but also for solving a large class of challenging problems in multidisciplinary fields of engineering, mathematics, and sciences”. Most of the papers dealing with (quadratically constrained) quadratic optimization problems and in which the results are obtained using CDT were published before 2010, but some of them are included in the recent book [9] (see [8,21,22]). Unfortunately, in almost all papers on CDT we are aware of there are unclear definitions, unconvincing
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Constantin Z˘alinescu [email protected]
1
Octav Mayer Institute of Mathematics, Ia¸si, Romania
2
Faculty of Mathematics, University “Al. I. Cuza” Ia¸si, Ia¸si, Romania
123
C. Z˘alinescu
proofs, and even false results. Examples supporting these claims are already mentioned in [28] and in our papers cited there. In this article we focus on quadratic optimization problems, and in Sect. 4 we give further examples supporting our claims. The aim of this paper is to rigorously treat quadratic optimization problems by the method suggested by CDT and to compare the results we obtain with the results on this class of problems obtained using CDT by Gao and his collaborators in various papers. In particular, we show that the so-called “complimentary-dual principle” is valid under very mild conditions, and does not depend on the type of constraints; for the min–max duality the strict positivity of the multipliers corresponding to nonlinear constraints is not necessary; all max–max and min–min duality relations are false.
2 Notations and preliminary results Let us consider the quadratic functions qk : Rn → R for k ∈ 0, m, that is qk (x) := 1 n n 2 x, Ak x − bk , x + ck for x ∈ R with Ak ∈ Sn , bk ∈ R (seen as column vector) a
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