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Convergence Analysis of Difference-of-Convex Algorithm with Subanalytic Data

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Abstract

Difference-of-Convex programming and related algorithms, which constitute the backbone of nonconvex programming and global optimization, were introduced in 1985 by Pham Dinh Tao and have been extensively developed by Le Thi Hoai An and Pham Dinh Tao since 1994 to become now classic and increasingly popular. That algorithm is a descent method without linesearch and every limit point of its generated sequence is a critical point of the related Difference-of-Convex program. Determining its convergence rate is a challenging problem. Its knowledge is crucial from both theoretical and practical points of view. In this work, we treat this problem for the class of Difference-of-Convex programs with subanalytic data by using the nonsmooth form of the Lojasiewicz inequality. We have successfully proved that the whole sequence is convergent, if it is bounded, provided that the objective function is subanalytic continuous on its domain and one of the two Difference-of-Convex components is differentiable with locally Lipschitz derivative. We also established a result on the convergence rate, which depended on the Lojasiewicz exponent of the objective function. Finally, for both classes of trust-region subproblems and nonconvex quadratic programs, we showed that the Lojasiewicz exponent was one half, and thereby, our proposed algorithms applied to these Difference-of-Convex programs were Root-linearly convergent.

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Acknowledgements

The authors would like to thank the referees and the Editor for their valuable comments and constructive suggestions, which have greatly helped to improve the quality and presentation of this paper.

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Authors and Affiliations

  1. Laboratory of Theoretical and Applied Computer Science, University of Lorraine, Site de Metz, Ile du Saulcy, 57045, Metz, France

    Hoai An Le Thi

  2. Department of Mathematics, University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam

    Van Ngai Huynh

  3. Laboratory of Mathematics, INSA-Rouen, University of Normandie, Avenue de l’Université, 76800, Saint-Étienne-du-Rouvray Cedex, France

    Tao Pham Dinh

Authors
  1. Hoai An Le Thi
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  2. Van Ngai Huynh
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  3. Tao Pham Dinh
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Correspondence to Hoai An Le Thi.

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The authors declare that they have no conflict of interest.

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Communicated by Jérôme Bolte.

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Le Thi, H.A., Huynh, V.N. & Pham Dinh, T. Convergence Analysis of Difference-of-Convex Algorithm with Subanalytic Data. J Optim Theory Appl 179, 103–126 (2018). https://doi.org/10.1007/s10957-018-1345-y

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  • DOI: https://doi.org/10.1007/s10957-018-1345-y

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