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Globally minimizing the sum of a convex–concave fraction and a convex function based on wave-curve bounds

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Abstract

We consider the problem of minimizing the sum of a convex–concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an \(\epsilon \)-approximate optimal solution in at most \(O\left( \frac{1}{\epsilon }\right) \) iterations. Numerical results demonstrate the efficiency of our algorithm.

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Acknowledgements

The authors are grateful to two anonymous referees for very helpful comments and suggestions.

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Authors and Affiliations

  1. LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing, 100191, People’s Republic of China

    Yong Xia

  2. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Longfei Wang

  3. School of Astronautics, Beihang University, Beijing, 100191, People’s Republic of China

    Xiaohui Wang

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  1. Yong Xia
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  2. Longfei Wang
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  3. Xiaohui Wang
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Correspondence to Xiaohui Wang.

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This research was supported by National Natural Science Foundation of China under Grants 11822103, 41804165, 11771056 and Beijing Natural Science Foundation Z180005.

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Xia, Y., Wang, L. & Wang, X. Globally minimizing the sum of a convex–concave fraction and a convex function based on wave-curve bounds. J Glob Optim 77, 301–318 (2020). https://doi.org/10.1007/s10898-019-00870-2

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  • DOI: https://doi.org/10.1007/s10898-019-00870-2

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