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Open issues and recent advances in DC programming and DCA

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  • Hoai An Le Thi

    (Université de Lorraine, LGIPM
    Institut Universitaire de France (IUF))

  • Tao Pham Dinh

    (University of Normandie)

Abstract

DC (difference of convex functions) programming and DC algorithm (DCA) are powerful tools for nonsmooth nonconvex optimization. This field was created in 1985 by Pham Dinh Tao in its preliminary state, then the intensive research of the authors of this paper has led to decisive developments since 1993, and has now become classic and increasingly popular worldwide. For 35 years from their birthday, these theoretical and algorithmic tools have been greatly enriched, thanks to a lot of their applications, by researchers and practitioners in the world, to model and solve nonconvex programs from many fields of applied sciences. This paper is devoted to key open issues, recent advances and trends in the development of these tools to meet the growing need for nonconvex programming and global optimization. We first give an outline in foundations of DC programming and DCA which permits us to highlight the philosophy of these tools, discuss key issues, formulate open problems, and bring relevant answers. After outlining key open issues that require deeper and more appropriate investigations, we will present recent advances and ongoing works in these issues. They turn around novel solution techniques in order to improve DCA’s efficiency and scalability, a new generation of algorithms beyond the standard framework of DC programming and DCA for large-dimensional DC programs and DC learning with Big data, as well as for broader classes of nonconvex problems beyond DC programs.

Suggested Citation

  • Hoai An Le Thi & Tao Pham Dinh, 2024. "Open issues and recent advances in DC programming and DCA," Journal of Global Optimization, Springer, vol. 88(3), pages 533-590, March.
    • Handle: RePEc:spr:jglopt:v:88:y:2024:i:3:d:10.1007_s10898-023-01272-1
    DOI: 10.1007/s10898-023-01272-1
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    References listed on IDEAS

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    1. M. V. Solodov, 2004. "On the Sequential Quadratically Constrained Quadratic Programming Methods," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 64-79, February.
    2. Hoai An Le Thi & Vinh Thanh Ho & Tao Pham Dinh, 2019. "A unified DC programming framework and efficient DCA based approaches for large scale batch reinforcement learning," Journal of Global Optimization, Springer, vol. 73(2), pages 279-310, February.
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    11. Hoai An Le Thi & Huynh Ngai & Tao Pham Dinh, 2016. "Error Bounds Via Exact Penalization with Applications to Concave and Quadratic Systems," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 228-250, October.
    12. Tao Pham Dinh & Van Ngai Huynh & Hoai An Le Thi & Vinh Thanh Ho, 2022. "Alternating DC algorithm for partial DC programming problems," Journal of Global Optimization, Springer, vol. 82(4), pages 897-928, April.
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