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DC programming and DCA: thirty years of developments

Abstract

The year 2015 marks the 30th birthday of DC (Difference of Convex functions) programming and DCA (DC Algorithms) which constitute the backbone of nonconvex programming and global optimization. In this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools. The survey is comprised of three parts. In the first part we present a brief history of the field, while in the second we summarize the state-of-the-art results and recent advances. We focus on main theoretical results and DCA solvers for important classes of difficult nonconvex optimization problems, and then give an overview of real-world applications whose solution methods are based on DCA. The third part is devoted to new trends and important open issues, as well as suggestions for future developments.

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Acknowledgements

The authors are grateful to Dr. Vo Xuan Thanh for sending us some references on DCA solvers for real-world applications, and the two anonymous reviewers as well as Professor Jong-Shi Pang for their constructive comments that greatly improved the manuscript, in particular one of reviewers for providing us some references on related DCA methods in Sect. 3.3.

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Correspondence to Hoai An Le Thi.

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Le Thi, H.A., Pham Dinh, T. DC programming and DCA: thirty years of developments. Math. Program. 169, 5–68 (2018). https://doi.org/10.1007/s10107-018-1235-y

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Keywords

  • DC programming
  • DCA
  • Theory
  • Algorithms
  • Applications

Mathematics Subject Classification

  • 90C26
  • 90C90