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On canonical duality theory and constrained optimization problems

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Abstract

Canonical duality theory (CDT) is presented by its creator DY Gao as a theory which can be used for solving a large class of challenging real-world problems. It is the aim of this paper to study rigorously constrained optimization problems in finite dimensional spaces using the method suggested by CDT and to discuss several results published in the last ten years.

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Notes

  1. The reference “[7]” is “Gao, D.Y.: Nonconvex semi-linear problems and canonical duality solutions, in Advances in Mechanics and Mathematics II. In: Gao, D.Y., Ogden R.W. (eds.), pp. 261–311. Kluwer Academic Publishers (2003)”.

  2. The reference “[26]” mentioned in [14, Rem. 1] is the item [7] from our bibliography, the others being the following: “8. Fang, S.C., Gao, D.Y., Sheu, R.L., Wu, S.Y.: Canonical dual approach to solving 0–1 quadratic programming problems. J. Ind. Manag. Optim. 4(4), 125–142 (2008)”, “12. Gao, D.Y.: Solutions and optimality criteria to box constrained nonconvex minimization problem. J. Ind. Manag. Optim. 3(2), 293–304 (2007)”, and “16. Gao, D.Y., Ruan, N.: Solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. 47, 463–484 (2010)”, respectively.

  3. The emphasized text can be found also in [3, p. NP26] and [4, p. 33]. One must also observe that for Latorre and Gao \(\mu \ne 0\) is equivalent to “\(\mu _{j}\ne 0\) \(\forall j=1,...,p\)”, and \((\lambda ,\mu )\in {\mathbb {R}}^{m\times p}\) if \(\lambda \in {\mathbb {R}}^{m}\) and \(\mu \in {\mathbb {R}}^{p}.\)

  4. It is worth quoting DY Gao’s comment from [1, p. 19] on our remark from [17, p. 1783] that the proof of [7, Th. 2] is not convincing: “Regarding the so-called “not convincing proof”, serious researcher should provide either a convincing proof or a disproof, rather than a complaint. Note that the canonical dual variables \(\sigma _{0}\) and \(\sigma _{1}\) are in two different levers (scales) with totally different physical units\(^{14}\), it is completely wrong to consider \((\sigma _{0},\sigma _{1})\) as one vector and to discuss the concavity of \(\varXi _{1}\left( x,(\cdot ,\cdot )\right) \) on \({\mathscr {S}}_{a} ^{+}\). The condition “\({\mathscr {S}}_{a}^{+}\) is convex” in Theorem 2 [5] should be understood in the way that \({\mathscr {S}}_{a}^{+}\) is convex in \(\sigma _{0}\) and \(\sigma _{1}\), respectively, as emphasized in Remark 1 [5]. Thus, the proof of Theorem 2 given in [5] is indeed convincing by simply using the classical saddle min-max duality for \((x,\sigma _{0})\) and \((x,\sigma _{1})\), respectively.”

    Note 14 from the text above is “Let us consider Example 1 in [5]. If the unit for x is the meter (m) and for q is Kg/m, then the units for the Lagrange multiplier \(\mu \) (dual to the constraint \(g(x)=\tfrac{1}{2}(\tfrac{1}{2}x^{2}-d)^{2}-e\)) should be \(Kg/m^{3}\) and for \(\sigma \) (canonical dual to \(\varLambda (x)=\tfrac{1}{2}x^{2}\)) should be Kg/m, respectively, so that each terms in \(\varXi _{1}(x,\mu ,\sigma )\) make physical sense”; “[5]” is our reference [7].

  5. Excepting [10], there are very few differences between the papers published in [2] and those having the same title from the retracted issue of the journal Mathematics and Mechanics of Solids dedicated to CDT.

  6. In fact, we did not find a correct proof of the “min-max” duality (like Eq. (20) from [10]) in DY Gao’s papers in the case \(Q_{0}\ne \overline{1,m}\).

  7. Referring to [9, Lem. 1], which is a reformulation of [15, Assertion II], the authors say: “The following Lemma is well known in mathematical programming (cf. Latorre and Gao [12] and Voisei and Zălinescu [13])”; the references “[12]” and “[13]” are our items [7] and [15], respectively. Of course, [9, Lem. 1] isFootnote 7 (continued)

    not “well known in mathematical programming”, being very specific to the problem considered in [6]. The reference [7] is not mentioned in [10] with respect to [9, Lem. 1].

  8. The reference “[3]” is the item [6] from our bibliography.

  9. The text is quoted from [10, p. 370]; here the reference “[15]” is “Wu, C., Gao, D.Y.: Canonical primal-dual method for solving nonconvex minimization problems. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Advances in Canonical Duality Theory. Springer, Berlin”. Note that the same text can be found in [8, p. 9] without any reference, as well as in [9, p. NP236], where the indicated reference is “Wu, C, Li, C, and Gao, DY. Canonical primal-dual method for solving nonconvex minimization problems. arXiv:1212.6492, 2012.” Observe that the main difference between arXiv:1212.6492 and reference “[15]” of [10] consists in the list of the authors, the content being practically the same.

References

  1. Gao, D.Y.: On unified modeling, canonical duality-triality theory, challenges and breakthrough in optimization, arXiv:1605.05534v3 (2016)

  2. Gao, D.Y., Latorre, V., Ruan, N.: Canonical Duality Theory Unified Methodology for Multidisciplinary Study. Advances in Mechanics and Mathematics, vol. 37. Springer, Cham (2017)

  3. Gao, D.Y., Ruan, N., Latorre, V.: RETRACTED: Canonical duality-triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex system, Math. Mech. Solids, 21(3), NP5–NP36 (2016) (see also arXiv:1410.2665)

  4. Gao, D.Y., Ruan, N., Latorre, V.: Canonical duality-triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex system. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Canonical Duality Theory. Advances in Mechanics and Mathematics, pp. 1–47. Springer, Cham (2017)

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  9. Morales-Silva, D., Gao, D.Y.: RETRACTED: on the minimal distance between two non-convex surfaces. Math. Mech. Solids 21(3), NP225–NP237 (2016). (compare with [8] and [10])

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Acknowledgements

We thank Prof. Marius Durea for reading a previous version of the paper and for his useful remarks. This work was presented at the 6th World Congress on Global Optimization, Metz, 2019; see [20] for an abridged version which contains the main results without proofs.

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  1. Octav Mayer Institute of Mathematics, Iasi Branch of Romanian Academy, 700505, Iaşi, Romania

    Constantin Zălinescu

  2. Faculty of Mathematics, University “Al. I. Cuza” Iaşi, 700506, Iaşi, Romania

    Constantin Zălinescu

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Zălinescu, C. On canonical duality theory and constrained optimization problems. J Glob Optim 82, 1053–1070 (2022). https://doi.org/10.1007/s10898-021-01021-2

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