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An Outcome-Space-Based Branch-and-Bound Algorithm for a Class of Sum-of-Fractions Problems

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Abstract

In this paper, we study the problem of maximizing the sum of a concave–convex quadratic fractional function on a non-empty, bounded, convex quadratic feasible set, which is called the problem (QCQFP). By using a conventional semidefinite program (SDP) relaxation, QCQFP is reformulated as a class of linear fractional programming problems over the cone of positive semidefinite matrices, which we refer to these problems as SDFP. After expatiating the properties of SDFP, we transform SDFP into its equivalent problem (ESDFP) whose non-convexity is mainly reflected in the newly added nonlinear equality constraints. By relaxing these nonlinear equality constraints into linear constraints, a special SDP relaxation is generated for ESDFP. Based on these results, an effective branch-and-bound algorithm is designed, and its theoretical convergence and worst-case complexity are proved. Numerical experiments demonstrate that the proposed algorithm is effective and feasible.

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Data availability statement

Supporting data for this study are available from the corresponding author, as reasonably requested.

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Acknowledgements

We thank each anonymous reviewer for their valuable comments and suggestions, which will help improve the quality of the papers.

Funding

This research is supported by the National Natural Science Foundation of China [Grant No. 11961001], the Construction Project of first-class subjects in Ningxia higher Education [Grant No. NXYLXK2017B09] and the Major proprietary funded project of North Minzu University [Grant No. ZDZX201901].

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, People’s Republic of China

    Bo Zhang & Xia Liu

  2. Ningxia Province Key Laboratory of Intelligent Information and Data Processing, North Minzu University, Yinchuan, 750021, People’s Republic of China

    YueLin Gao

  3. School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, People’s Republic of China

    YueLin Gao & XiaoLi Huang

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  1. Bo Zhang
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  2. YueLin Gao
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  3. Xia Liu
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  4. XiaoLi Huang
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Correspondence to YueLin Gao.

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Communicated by Miguel F. Anjos.

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Zhang, B., Gao, Y., Liu, X. et al. An Outcome-Space-Based Branch-and-Bound Algorithm for a Class of Sum-of-Fractions Problems. J Optim Theory Appl 192, 830–855 (2022). https://doi.org/10.1007/s10957-021-01992-y

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  • DOI: https://doi.org/10.1007/s10957-021-01992-y

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