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An Outer Space Approach to Tackle Generalized Affine Fractional Program Problems

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Abstract

This paper aims to globally solve a generalized affine fractional program problem (GAFPP). Firstly, by introducing some outer space variables and performing equivalent transformations, we can derive the equivalence problem (EP) of the GAFPP. Secondly, by constructing a novel linear relaxation method, we can deduce the affine relaxation problem (ARP) of the EP. Next, by solving the ARP to compute the lower bound, we propose a new outer space branch-and-bound algorithm for tackling the GAFPP. Then, the global convergence of the algorithm is proved, and the computational complexity of the algorithm in the worst case is analyzed. Finally, numerical experimental results are reported to illustrate the effectiveness of the algorithm.

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Data Availability Statement

The data supporting this study can be obtained based on reasonable requests from the corresponding author.

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Acknowledgements

The authors are very grateful to the anonymous referees for their helpful suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11871196, 12071133, 12071112), China Postdoctoral Science Foundation (2017M622340), the Key Scientific and Technological Research Projects in Henan Province (232102211085, 202102210147).

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Authors and Affiliations

  1. School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang, 453003, China

    Hongwei Jiao & Binbin Li

  2. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, China

    Youlin Shang

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  1. Hongwei Jiao
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  2. Binbin Li
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Correspondence to Hongwei Jiao.

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Appendix

Appendix

Example A1 (Shen et al. [32])

$$\begin{aligned} {\left\{ \begin{array}{ll} \min &{} \dfrac{-x_{1}+2 x_{2}+2}{3 x_{1}-4 x_{2}+5} \times \dfrac{4 x_{1}-3 x_{2}+4}{-2 x_{1}+x_{2}+3}, \\ \text{ s.t. } &{} x_{1}+x_{2} \leqslant 1.5, \\ &{} x_{1} - x_{2} \leqslant 0, \\ &{} 0 \leqslant x_{1} \leqslant 1, \ 0 \leqslant x_{2} \leqslant 1. \end{array}\right. } \end{aligned}$$

Example A2 (Jiao [9])

$$\begin{aligned} {\left\{ \begin{array}{ll} \max &{} \dfrac{3 x_{1}+5 x_{2}+3 x_{3}+50}{3 x_{1}+4 x_{2} + 5 x_{3}+50}+\dfrac{3 x_{1} + 4 x_{2}+50}{4 x_{1}+ 3 x_{2} + 2 x_{3}+50}+\dfrac{4 x_{1}+2 x_{2}+4 x_{3}+50}{5 x_{1}+4 x_{2}+3 x_{3}+50}, \\ \text { s.t. } &{} 6 x_{1}+3 x_{2}+3 x_{3} \leqslant 10, \\ &{} 10 x_{1}+3 x_{2}+8 x_{3} \leqslant 10, \\ &{} 0 \leqslant x_{1} \leqslant 1, \ 0 \leqslant x_{2} \leqslant 3.333333, \ 0 \leqslant x_{3} \leqslant 1. \end{array}\right. } \end{aligned}$$

Example A3 (Jiao et al. [10, 15])

$$\begin{aligned} {\left\{ \begin{array}{ll} \min &{} \left( \dfrac{x_{1}+x_{2}+1}{x_{1} + x_{2} + 2} \right) ^{1.5} \times \left( \dfrac{x_{1}+x_{2}+3}{x_{1}+x_{2}+4} \right) ^{2.1} \times \left( \dfrac{x_{1}+x_{2}+5}{x_{1}+x_{2}+6} \right) ^{1.2} \times \left( \dfrac{x_{1}+x_{2}+7}{x_{1}+x_{2}+8} \right) ^{1.1}, \\ \text{ s.t. } \ &{} x_{1}=x_{2},\\ &{} 1.0 \leqslant x_{1} \leqslant 2.0, \ 1.0 \leqslant x_{2} \leqslant 2.0. \end{array}\right. } \end{aligned}$$

Example A4 (Pei and Zhu [28])

$$\begin{aligned} {\left\{ \begin{array}{ll} \max &{} \dfrac{3 x_{1}+5 x_{2}+3 x_{3}+50}{3 x_{1}+4 x_{2}+5 x_{3}+50}+\dfrac{3 x_{1}+5 x_{2}+50}{3 x_{1}+ 5 x_{2}+ 3 x_{3}+50}+\dfrac{4 x_{1}+2 x_{2}+4 x_{3}+50}{5 x_{1}+4 x_{2}+3 x_{3}+50}, \\ \text{ s.t. } &{} 6 x_{1}+3 x_{2}+3 x_{3} \leqslant 10, \\ &{} 10 x_{1}+3 x_{2}+8 x_{3} \leqslant 10, \\ &{} x_{1} \geqslant 0, \ x_{2} \geqslant 0, \ x_{3} \geqslant 0. \end{array}\right. } \end{aligned}$$

Example A5 (Pei and Zhu [28])

$$\begin{aligned} {\left\{ \begin{array}{ll} \max &{} \dfrac{4 x_{1} + 3 x_{2} + 3 x_{3}+50}{3 x_{2} + 3 x_{3}+50} + \dfrac{3 x_{1} + 4 x_{2}+50}{4 x_{1}+ 4 x_{2}+ 5 x_{3}+50} + \dfrac{x_{1} + 2 x_{2} + 5 x_{3}+50}{x_{1}+ 5 x_{2} + 5 x_{3}+50} \\ &{} +\dfrac{x_{1} + 2 x_{2} + 4 x_{3}+50}{5 x_{2} + 4 x_{3}+50}, \\ \text{ s.t. } &{} 2 x_{1} + x_{2} + 5 x_{3} \leqslant 10, \\ &{} x_{1} + 6 x_{2} + 3 x_{3} \leqslant 10,\\ &{} 5 x_{1} + 9 x_{2} + 2 x_{3} \leqslant 10, \\ &{} 9 x_{1} + 7 x_{2} + 3 x_{3} \leqslant 10, \\ &{} x_{1} \geqslant 0, \ x_{2} \geqslant 0, \ x_{3} \geqslant 0. \end{array}\right. } \end{aligned}$$

Example A6 (Shen and Lu [35])

$$\begin{aligned} {\left\{ \begin{array}{ll} \min &{} \dfrac{37 x_{1} + 73 x_{2} + 13}{13 x_{1} + 13 x_{2} + 13} + \dfrac{63 x_{1} - 18 x_{2} + 39 }{13 x_{1} + 26x_{2} + 13}, \\ \text { s.t. } &{} 5 x_{1}- 3 x_{2} = 3, \\ &{} 1.5 \leqslant x_{1} \leqslant 3. \end{array}\right. } \end{aligned}$$

Example A7 (Wang and Shen [44])

$$\begin{aligned} {\left\{ \begin{array}{ll} \min &{} \dfrac{4 x_{1} + 3 x_{2} + 3 x_{3}+50}{3 x_{2} + 3 x_{3}+50} + \dfrac{3 x_{1} + 4 x_{3}+50}{4 x_{1}+ 4 x_{2}+ 5 x_{3}+50} + \dfrac{x_{1} + 2 x_{2} + 4 x_{3}+50}{x_{1}+ 5 x_{2} + 5 x_{3}+50} \\ &{} +\dfrac{x_{1} + 2 x_{2} + 4 x_{3}+50}{5 x_{2} + 4 x_{3}+50}, \\ \text{ s.t. } &{} 2 x_{1} + x_{2} + 5 x_{3} \leqslant 10, \\ &{} x_{1} + 6 x_{2} + 2 x_{3} \leqslant 10,\\ &{} 9 x_{1} + 7 x_{2} + 3 x_{3} \geqslant 10, \\ &{} x_{1} \geqslant 0, \ x_{2} \geqslant 0, \ x_{3} \geqslant 0. \end{array}\right. } \end{aligned}$$

Example A8 (Shen et al. [32])

$$\begin{aligned} {\left\{ \begin{array}{ll} \min &{} \prod \limits _{i=1}^{6} \dfrac{\left\langle c_{i}, x\right\rangle + s_{i}}{\left\langle d_{i}, x\right\rangle + r_{i}}, \\ \text { s.t. } &{} A x \leqslant b, x \geqslant 0, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned}{} & {} \begin{array}{rrrrrrrrrrrrr} c_{1}=&{} (-0.2 &{} -0.7 &{} -0.1 &{} 0.4 &{} 0.0 &{} 0.8 &{} 0.1 &{} -0.8 &{} -0.2 &{} 0.0 &{} 0.1 &{} 0.4), \\ d_{1}=&{} (0.2 &{} 0.5 &{} -0.6 &{} 0.1 &{} 0.6 &{} 0.4 &{} -0.4 &{} -0.3 &{} 0.7 &{} 0.5 &{} 0.4 &{} -0.1), \\ c_{2}=&{} (-0.1 &{} 0.1 &{} -0.4 &{} -0.1 &{} -0.1 &{} 0.4 &{} 0.2 &{} 0.5 &{} 0.3 &{} -0.4 &{} -0.3 &{} 0.3), \\ d_{2}=&{} (-0.3 &{} -0.2 &{} -0.7 &{} 0.1 &{} 0.2 &{} -0.2 &{} -0.5 &{} 0.4 &{} 0.3 &{} 0.0 &{} 0.6 &{} -0.5), \\ c_{3}=&{} (0.8 &{} 0.0 &{} -0.1 &{} 0.4 &{} 0.2 &{} 0.1 &{} -0.5 &{} 0.0 &{} 0.5 &{} 0.6 &{} -0.3 &{} -0.4), \\ d_{3}=&{} (0.1 &{} 0.0 &{} 0.0 &{} 0.3 &{} 0.2 &{} 0.7 &{} 0.4 &{} 0.2 &{} -0.1 &{} -0.5 &{} 0.6 &{} -0.1), \\ c_{4}=&{} (0.6 &{} 0.2 &{} 0.2 &{} -0.3 &{} 0.5 &{} 0.4 &{} 0.1 &{} 0.6 &{} -0.3 &{} 0.3 &{} 0.4 &{} 0.3), \\ d_{4}=&{} (-0.3 &{} 0.0 &{} 0.0 &{} -0.5 &{} -0.1 &{} 0.2 &{} 0.6 &{} -0.6 &{} 0.1 &{} -0.2 &{} 0.8 &{} -0.3), \\ c_{5}=&{} (-0.3 &{} -0.3 &{} 0.5 &{} 0.1 &{} 0.2 &{} -0.5 &{} 0.1 &{} 0.2 &{} 0.0 &{} 0.6 &{} 0.3 &{} -0.2), \\ d_{5}=&{} (0.3 &{} 0.0 &{} 0.3 &{} 0.0 &{} -0.8 &{} -0.3 &{} 0.3 &{} -0.9 &{} -0.1 &{} -0.6 &{} -0.1 &{} 0.2), \\ c_{6}=&{} (0.2 &{} -0.1 &{} 0.0 &{} 0.0 &{} -0.2 &{} -0.4 &{} 0.0 &{} -0.6 &{} 0.8 &{} -0.2 &{} 0.0 &{} -0.1), \\ d_{6}=&{} (0.0 &{} 0.6 &{} 0.0 &{} 0.1 &{} 0.0 &{} -0.2 &{} 0.0 &{} -0.5 &{} 0.2 &{} -0.3 &{} 0.3 &{} 0.1), \end{array} \end{aligned}$$
$$\begin{aligned}{} & {} \begin{aligned} A=\left[ \begin{array}{rrrrrrrrrrrr} 1.9 &{} 0.0 &{} -0.2 &{} -1.5 &{} 1.8 &{} 0.9 &{} -1.0 &{} 4.5 &{} 4.5 &{} -3.5 &{} -1.8 &{} -4.8 \\ 2.9 &{} 3.7 &{} -4.8 &{} -1.9 &{} 1.8 &{} -3.7 &{} 1.8 &{} 2.5 &{} -2.9 &{} 1.9 &{} -3 &{} 3.2 \\ 3.3 &{} 2.4 &{} 3.3 &{} 4.8 &{} -0.3 &{} 3.9 &{} 0.8 &{} -1.7 &{} 2.0 &{} -0.3 &{} -1.8 &{} 2.2 \\ -4.3 &{} 1.8 &{} 2.1 &{} -4.5 &{} -0.5 &{} 2.4 &{} 1.4 &{} -0.3 &{} -2.0 &{} -2.8 &{} 0.4 &{} 4.5 \\ 1.5 &{} -0.3 &{} 0.4 &{} 1.2 &{} 1.1 &{} 1.9 &{} 1.5 &{} -1.2 &{} -3.3 &{} 4.4 &{} 3.2 &{} -4.3 \\ -3.2 &{} 2.4 &{} -4.5 &{} -1.0 &{} -2.7 &{} 3.7 &{} -0.1 &{} 3.9 &{} -1.9 &{} 3.2 &{} 2.1 &{} 1.3 \\ 0.9 &{} 0.5 &{} 4.0 &{} -1.5 &{} 1.2 &{} -1.5 &{} 1.2 &{} -3.7 &{} -0.1 &{} 0.0 &{} -2.4 &{} -4.1 \\ -4.1 &{} -4.5 &{} 2.2 &{} -3.1 &{} 4.4 &{} 4.8 &{} -3.4 &{} 2.2 &{} -2.1 &{} 2.3 &{} 2.6 &{} -1.4 \\ 2.4 &{} 2.3 &{} 4.7 &{} -1.7 &{} -1.6 &{} 3.8 &{} -4.0 &{} 1.3 &{} -0.4 &{} -0.4 &{} 2.9 &{} 1.2 \\ 0.0 &{} -3.2 &{} -0.2 &{} 2.0 &{} -2.9 &{} 2.7 &{} 3.1 &{} 2.9 &{} -2.6 &{} -4.3 &{} 0.2 &{} 4.6 \\ -1.3 &{} -0.9 &{} 3.4 &{} 3.9 &{} 4.9 &{} 2.3 &{} -3.0 &{} -1.5 &{} 2.5 &{} -1.7 &{} 1.7 &{} -2.9 \\ 3.5 &{} 3.4 &{} 2.5 &{} -0.4 &{} -4.5 &{} 2.8 &{} -1.7 &{} 2.1 &{} -2.9 &{} -4.7 &{} 1.3 &{} 4.5 \\ 1.9 &{} -0.9 &{} -3.3 &{} -2.3 &{} 1.6 &{} -0.5 &{} -4.9 &{} 3.0 &{} -4.9 &{} 3.6 &{} -3.7 &{} 2.2 \\ -1.4 &{} 3.5 &{} -2.8 &{} -1.2 &{} -4.7 &{} -3.2 &{} 2.2 &{} -4.0 &{} 2.8 &{} 3.3 &{} 4.4 &{} -3.1 \\ -2.1 &{} 2.6 &{} -3.9 &{} 1.0 &{} 2.3 &{} 1.8 &{} 4.2 &{} 1.8 &{} 2.7 &{} 0.9 &{} 3.3 &{} 1.7 \end{array} \right] , \end{aligned} \\{} & {} s=(21,16.3, 3.7, -1.8, 5, 12.7)^{\top }, r=(13.3,16, 16.7, 21.5, 18.7, 19.2)^{\top },\\{} & {} b=(-20.1,-1.0, 82.6, 14.6, 37.7, 40.7,\\{} & {} \qquad -23, 47.4, 83.0, 9.9, 33.7, 49.1, 14.0,-45.6, 30.4)^{\top }. \end{aligned}$$

Example A9 (Shen and Jiao [34])

$$\begin{aligned} {\left\{ \begin{array}{ll} \min &{} 169 \times \dfrac{x_{1} + x_{2}+1}{37 x_{1} + 73 x_{2}+13 } \times \dfrac{x_{1} + 2 x_{2} + 1}{63 x_{1} - 18 x_{2} + 39}, \\ \text { s.t. } &{} 5x_{1} - 3 x_{2} \leqslant 3, \\ &{} 1.5 \leqslant x_{1} \leqslant 3, \ 2 \leqslant x_{2} \leqslant 3.5. \end{array}\right. } \end{aligned}$$

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Jiao, H., Li, B. & Shang, Y. An Outer Space Approach to Tackle Generalized Affine Fractional Program Problems. J Optim Theory Appl 201, 1–35 (2024). https://doi.org/10.1007/s10957-023-02368-0

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