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DC Optimization Problems

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Convex Analysis and Global Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 110))

Abstract

This chapter presents methods and algorithms for solving the basic dc optimization problems: concave minimization under linear constraints (Sect. 7.1), concave minimization under convex constraints (Sect. 7.2), reverse convex programming (Sect. 7.3), general canonical dc optimization problem (Sect. 7.4), general robust approach to dc optimization (Sect. 7.5), and also applications of dc optimization in various fields (Sects. 7.6–7.8) such as design calculations, location, distance geometry, and clustering. An important new result in this chapter is a surprisingly simple proof of the convergence of algorithms using ω-subdivision—a question which had remained open during two decades before being settled with quite sophisticated proofs by Locatelli (Math Program 85:593–616, 1999), Jaumard and Meyer (J Optim Theory Appl 110:119–144, 2001), and more recently Kuno and Ishihama (J Glob Optim 61:203–220, 2015). In addition, the new concept of ω-bisection by Kuno–Ishihama is introduced as a substitute for ω-subdivision.

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Notes

  1. 1.

    In the above formulation the function h(x) is assumed to be finite throughout \(\mathbb{R}^{n};\) if this condition may not be satisfied (as in the last example) certain results below must be applied with caution.

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Authors and Affiliations

  1. Vietnam Academy of Science and Technology, Institute of Mathematics, Hanoi, Vietnam

    Hoang Tuy

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  1. Hoang Tuy
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Tuy, H. (2016). DC Optimization Problems. In: Convex Analysis and Global Optimization. Springer Optimization and Its Applications, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-31484-6_7

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