To test the proposed algorithm in efficiency and solution quality, we performed some computational examples on a personal computer containing an Intel Core i5 processor of 2.40 GHz and 4GB of RAM. The code base is written in Matlab 2014a and interfaces LINPROG for the linear relaxation subproblems and CVX for the convex relaxation subproblems.

We consider some numerical examples in recent literatures [14, 20, 21, 22, 23, 24, 25, 26], and a randomly generated test problem to verify the performance of the algorithm. The numerical test and results are listed as follows.

#### Example 5.1

([23]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{-{x}_{1}+2{x}_{2}+2}{3{x}_{1}-4{x}_{2}+5}+\frac{4{x}_{1}-3{x}_{2}+4}{-2{x}_{1}+{x}_{2}+3}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}{x}_{1}+{x}_{2}\le 1.50,{x}_{1}-{x}_{2}\le 0,0\le {x}_{1}\le 1,0\le {x}_{2}\le 1.\end{array}}\end{array}$$

#### Example 5.2

([14, 22, 23]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{4{x}_{1}+3{x}_{2}+3{x}_{3}+50}{3{x}_{2}+3{x}_{3}+50}+\frac{3{x}_{1}+4{x}_{2}+50}{4{x}_{1}+4{x}_{2}+5{x}_{3}+50}+\frac{{x}_{1}+2{x}_{2}+5{x}_{3}+50}{{x}_{1}+5{x}_{2}+5{x}_{3}+50}+\frac{{x}_{1}+2{x}_{2}+4{x}_{3}+50}{5{x}_{2}+4{x}_{3}+50}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}2{x}_{1}+{x}_{2}+5{x}_{3}\le 10,{x}_{1}+6{x}_{2}+3{x}_{3}\le 10,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}5{x}_{1}+9{x}_{2}+2{x}_{3}\le 10,9{x}_{1}+7{x}_{2}+3{x}_{3}\le 10,{x}_{1}\ge 0,{x}_{2}\ge 0,{x}_{3}\ge 0.\end{array}}\end{array}$$

#### Example 5.3

([14]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}0.9\times \frac{-{x}_{1}+2{x}_{2}+2}{3{x}_{1}-4{x}_{2}+5}-0.1\times \frac{4{x}_{1}-3{x}_{2}+4}{-2{x}_{1}+{x}_{2}+3}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}{x}_{1}+{x}_{2}\le 1.50,{x}_{1}-{x}_{2}\le 0,0\le {x}_{1}\le 1,0\le {x}_{2}\le 1.\end{array}}\end{array}$$

#### Example 5.4

([21]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{3{x}_{1}+4{x}_{2}+50}{3{x}_{1}+5{x}_{2}+4{x}_{3}+50}-\frac{3{x}_{1}+5{x}_{2}+3{x}_{3}+50}{5{x}_{1}+5{x}_{2}+4{x}_{3}+50}-\frac{{x}_{1}+2{x}_{2}+4{x}_{3}+50}{5{x}_{2}+4{x}_{3}+50}-\frac{4{x}_{1}+3{x}_{2}+3{x}_{3}+50}{3{x}_{2}+3{x}_{3}+50}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}6{x}_{1}+3{x}_{2}+3{x}_{3}\le 10,10{x}_{1}+3{x}_{2}+8{x}_{3}\le 10,{x}_{1}\ge 0,{x}_{2}\ge 0,{x}_{3}\ge 0.\end{array}}\end{array}$$

#### Example 5.5

([14, 24]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{37{x}_{1}+73{x}_{2}+13}{13{x}_{1}+13{x}_{2}+13}+\frac{63{x}_{1}-18{x}_{2}+39}{13{x}_{1}+26{x}_{2}+13}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}5{x}_{1}-3{x}_{2}=3,1.5\le {x}_{1}\le 3.\end{array}}\end{array}$$

#### Example 5.6

([20, 23]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{3{x}_{1}+5{x}_{2}+3{x}_{3}+50}{3{x}_{1}+4{x}_{2}+5{x}_{3}+50}+\frac{3{x}_{1}+4{x}_{2}+50}{4{x}_{1}+3{x}_{2}+2{x}_{3}+50}+\frac{4{x}_{1}+2{x}_{2}+4{x}_{3}+50}{5{x}_{1}+4{x}_{2}+3{x}_{3}+50}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}6{x}_{1}+3{x}_{2}+3{x}_{3}\le 10,10{x}_{1}+3{x}_{2}+8{x}_{3}\le 10,{x}_{1}\ge 0,{x}_{2}\ge 0,{x}_{3}\ge 0.\end{array}}\end{array}$$

#### Example 5.7

([14, 24]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{i=1}^{5}\frac{{c}^{i}x+{r}_{i}}{{d}^{i}x+{s}_{i}}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}Ax\le b,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x\ge 0.\end{array}}\end{array}$$

*where*

$$\begin{array}{}{\displaystyle b=(15.7,31.8,-36.4,38.5,40.3,10.0,89.8,5.8,2.7,-16.3,-14.6,-72.7,57.7,-34.5,69.1{)}^{T}}\end{array}$$

$$\begin{array}{}{\displaystyle A=\left(\begin{array}{llllllllllll}-1.8& -2.2& 0.8& 4.1& 3.8& -2.3& -0.8& 2.5& -1.6& 0.2& -4.5& -1.8\\ 4.6& -2.0& 1.4& 3.2& -4.2& -3.3& 1.9& 0.7& 0.8& -4.4& 4.4& 2.0\\ 3.7& -2.8& -3.2& -2.0& -3.7& -3.3& 3.5& -0.7& 1.5& -3.1& 4.5& -1.1\\ -0.6& -0.6& -2.5& 4.1& 0.6& 3.3& 2.8& -0.1& 4.1& -3.2& -1.2& -4.3\\ 1.8& -1.6& -4.5& -1.3& 4.6& 3.3& 4.2& -1.2& 1.9& 2.4& 3.4& -2.9\\ -0.5& -4.1& 1.7& 3.9& -0.1& -3.9& -1.5& 1.6& 2.3& -2.3& -3.2& 3.9\\ 0.3& 1.7& 1.3& 4.7& 0.9& 3.9& -0.5& -1.2& 3.8& 0.6& -0.2& -1.5\\ 0.5& -4.2& 3.6& -0.6& -4.8& 1.5& -0.3& 0.6& -3.6& 0.2& 3.8& -2.8\\ 0.1& 3.3& -4.3& 2.4& 4.1& 1.7& 1.0& -3.3& 4.4& -3.7& -1.1& -1.4\\ -0.6& 2.2& 2.5& 1.3& -4.3& -2.9& -4.1& 2.7& -0.8& -2.9& 3.5& 1.2\\ 4.3& 1.9& -4.0& -2.6& 1.8& 2.5& 0.6& 1.3& -4.3& -2.3& 4.1& -1.1\\ 0.0& 0.4& -4.5& -4.4& 1.2& -3.8& -1.9& 1.2& 3.0& -1.1& -0.2& 2.5\\ -0.1& -1.7& 2.9& 1.5& 4.7& -0.3& 4.2& -4.4& -3.9& 4.4& 4.7& -1.0\\ -3.8& 1.4& -4.7& 1.9& 3.8& 3.5& 1.5& 2.3& -3.7& -4.2& 2.7& -0.1\\ 0.2& -0.1& 4.9& -0.9& 0.1& 4.3& 1.6& 2.6& 1.5& -1.0& 0.8& 1.6\end{array}\right)}\end{array}$$

$$\begin{array}{}\begin{array}{lr}{c}^{1}=(0.0,-0.1,-0.3,0.3,0.5,0.5,-0.8,0.4,-0.4,0.2,0.2,-0.1),& {r}_{1}=14.6\\ {c}^{2}=(0.2,0.5,0.0,0.4,0.1,-0.6,-0.1,-0.2,-0.2,0.1,0.2,0.3),& {r}_{2}=7.1\\ {c}^{3}=(-0.1,0.3,0.0,0.1-0.1,0.0,0.3,-0.2,0.0,0.3,0.5,0.3),& {r}_{3}=1.7\\ {c}^{4}=(-0.1,0.5,0.1,0.1-0.2,-0.5,0.6,0.7,0.5,0.7,-0.1,0.1),& {r}_{4}=4.0\\ {c}^{5}=(0.7,-0.5,0.1,0.2-0.1,-0.3,0.0,-0.1,-0.2,0.6,0.5,-0.2),& {r}_{5}=6.8\\ {d}^{1}=(-0.3,-0.1,-0.1,-0.1,0.1,0.4,0.2,-0.2,0.4,0.2,-0.4,0.3),& {s}_{1}=14.2\\ {d}^{2}=(0.0,0.1,-0.1,0.3,0.3-0.2,0.3,0.0,-0.4,0.5,-0.3,0.1),& {s}_{2}=1.7\\ {d}^{3}=(0.8,-0.4,0.7,-0.4,-0.4,0.5,-0.2,-0.8,0.5,0.6,-0.2,0.6),& {s}_{3}=8.1\\ {d}^{4}=(0.0,0.6,-0.3,0.3,0.0,0.2,0.3,-0.6,-0.2,-0.5,0.8,-0.5),& {s}_{4}=26.9\\ {d}^{5}=(0.4,0.2,-0.2,0.9,0.5,-0.1,0.3,-0.8,-0.2,0.6,-0.2,-0.4),& {s}_{5}=3.7\end{array}\end{array}$$

#### Example 5.8

([25]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}min\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{1}+{x}_{2}+{x}_{3}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{{\displaystyle 833.33252{x}_{4}}}{{\displaystyle {x}_{1}{x}_{6}}}+\frac{{\displaystyle 100}}{{\displaystyle {x}_{6}}}\le 1\\ \phantom{\rule{2em}{0ex}}\frac{{\displaystyle 1250{x}_{5}-1250{x}_{4}}}{{\displaystyle {x}_{2}{x}_{7}}}+\frac{{\displaystyle {x}_{4}}}{{\displaystyle {x}_{7}}}\le 1\\ \phantom{\rule{2em}{0ex}}\frac{{\displaystyle 1250000-2500x5}}{{\displaystyle {x}_{3}{x}_{8}}}+\frac{{\displaystyle {x}_{5}}}{{\displaystyle {x}_{8}}}\le 1\\ \phantom{\rule{2em}{0ex}}0.0025{x}_{4}+0.0025{x}_{6}\le 1\\ \phantom{\rule{2em}{0ex}}-0.0025{x}_{4}+0.0025{x}_{5}+0.0025{x}_{7}\le 1\\ \phantom{\rule{2em}{0ex}}0.01{x}_{8}-0.01{x}_{5}\le 1\\ \phantom{\rule{2em}{0ex}}100\le {x}_{1}\le 10000\\ \phantom{\rule{2em}{0ex}}1000\le {x}_{2},{x}_{3}\le 10000\\ \phantom{\rule{2em}{0ex}}10\le {x}_{i}\le 1000,i=4,5\dots ,8.\end{array}}\end{array}$$

#### Example 5.9

([26]).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}min\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{{\displaystyle 0.5({x}_{1}-10)}}{{\displaystyle {x}_{2}}}-{x}_{1}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{{\displaystyle {x}_{2}}}{{\displaystyle {x}_{3}}}+{x}_{1}+0.5{x}_{1}{x}_{3}\le 100\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\le {x}_{i}\le 100,\phantom{\rule{thinmathspace}{0ex}}i=1,2,3.\end{array}}\end{array}$$

#### Example 5.10

(Random test).

$$\begin{array}{}{\displaystyle \{\begin{array}{l}max\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{i=1}^{p}{\delta}_{i}\frac{({n}^{i}{)}^{T}x+{\beta}_{i}}{({d}^{i}{)}^{T}x+{\gamma}_{i}}\\ \mathrm{s}.\mathrm{t}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}Ax\le b,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x\ge 0.\end{array}}\end{array}$$

*where the elements of the matrix* *A* ∈ *R*_{m×n}, *b* ∈ *R*^{m}, *n*^{i}, *d*^{i} ∈ *R*^{n} *and the elements of constant terms of denominators and numerators* *β*_{i} *and* *γ*_{i} ∈ *R* *are randomly generated in the interval [0, 1], this agrees with the way random numbers are generated in [14], while in our experiment* *δ*_{i} ∈ *R* *is randomly generated in the interval* *[*−*1, 1] rather than in interval [0, 1], this is much more challenging to test the performance of the algorithm*. *The results of the contrast experiments and the random tests are shown in *-*, and each symbol used in the table has the following meaning* *p*, *m* *and* *n* *represent the number of affine ratios in the objective function, the number of the constraints and the number of constrained variable respectively; Ave*.*time, Ave*.*Nod and Ave*.*Ite stand for the average CPU time in seconds, average number of the subproblem and iteration in the algorithm;* *ϵ* *express the error precision used in the algorithm, and* *α* *refer to the split ratio used in the branching operations*.

Table 1 Results of the numerical contrast test 1-7.

Table 2 Computational results of random test 8 corresponding to the variation of the number of variable *n*.

Table 3 Computational results of random test 8 corresponding to the variation of the number of ratio *p*.

while

$$\begin{array}{}{\displaystyle {x}^{(18)}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}(6.24409,20.0249,3.79672,5.93972,0,7.43852,}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}0,23.2833,0.515015,40.9896,0,3.14363{)}^{T}\\ {x}^{(22)}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}(6.223689,20.060317,3.774684,5.947841,0,7.456686,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}0,23.312579,0.000204,41.031824,0,3.171106{)}^{T}\\ {x}^{(23)}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}(578.973143,1359.572730,5110.701048,181.9898,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}295.5719,218.0101,286.4179,395.5719)\\ {x}^{(24)}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}(87.614446,8.754375,1.413643,19.311410)\\ {x}^{(our1)}=(6.22442,20.05821,3.77441,5.94859,0.00001,7.45691,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}0.00002,23.31133,0.00012,41.03002,0.00001,3.17225{)}^{T}\\ {x}^{(our2)}=(579.326059,1359.9445,5109.977472,182.019317,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}295.600901,217.980682,286.418416,395.600901)\\ {x}^{(our3)}=(87.614446,8.754375,1.413643,19.311410).\end{array}$$

The computational results in and indicate that our algorithm has good performance, and is effective for special relatively large-scale optimization problems where the number of ratios in the objective function is not so large. Meanwhile, we find that, the average number of iterations and subproblems that need to be solved by the algorithm and the average CPU time do not substantially increase as the size of the problem becomes large. Based on the result of the above numerical examples, our algorithm is quite robust and efficient and so it can be used successfully to solve the sum of affine ratios problem SRP.

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