Global algorithm for effectively solving min-max affine fractional programs

Abstract

This article investigates solving the min-max affine fractional programming problem (MAFPP), which arises in systems science and engineering. For globally solving the MAFPP, based on the outer space branching search, we design a branch-relaxation-bound algorithm. In the algorithm, we firstly convert the MAFPP into an equivalent problem with linear fractional constraints by introducing some auxiliary parameter variables. Next, to determine and update the lower bound during the branching search process, we construct the linear relaxation problem of the equivalence problem by using a new relaxation technique. The global convergence of the presented algorithm is verified. Also, by analyzing the algorithmic complexity, we give a maximum estimate of iteration times of the presented algorithm. Finally, computational comparison results are reported to demonstrate the effectiveness and feasibility of the presented algorithm.

Appendix A

A.1 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \min \max \displaystyle \left\{ \frac{3y_{1}+y_{2}-2y_{3}+0.8}{2y_{1}-y_{2}+y_{3}},\frac{4y_{1}-2y_{2}+y_{3}}{7y_{1}+3y_{2}-y_{3}}\right\} \\ {\mathrm{s. t.}}\ \ y_{1}+y_{2}-y_{3}\le 1,\\ \ \ \ \ \ \ -y_{1}+y_{2}-y_{3}\le -1,\\ \ \ \ \ \ \ 12y_{1}+5y_{2}+12y_{3}\le 34.8,\\ \ \ \ \ \ \ 12y_{1}+12y_{2}+7y_{3}\le 29.1,\\ \ \ \ \ \ \ \ -6y_{1}+y_{2}+y_{3}\le -4.1,\\ \ \ \ \ \ \ \ 1.0\le y_{1}\le 1.1,\ 0.55\le y_{2}\le 0.65,\ 1.35\le y_{3}\le 1.45. \end{array} \right. \end{aligned}$$

A.2 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \max \min \displaystyle \left\{ \frac{37y_{1}+73y_{2}+13}{13y_{1}+13y_{2}+13},\frac{63y_{1}-18y_{2}+39}{13y_{1}+26y_{2}+13}\right\} \\ {\mathrm{s. t.}}\ \ 5y_{1}-3y_{2}=3, \\ \ \ \ \ \ \ 1.5\le y_{1}\le 3. \end{array} \right. \end{aligned}$$

A.3 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \min \max \displaystyle \left\{ \frac{2y_{1}+2y_{2}-y_{3}+0.9}{y_{1}-y_{2}+y_{3}},\frac{3y_{1}-y_{2}+y_{3}}{8y_{1}+4y_{2}-y_{3}}\right\} \\ {\mathrm{s. t.}}\ \ y_{1}+y_{2}-y_{3}\le 1,\\ \ \ \ \ \ \ -y_{1}+y_{2}-y_{3}\le -1,\\ \ \ \ \ \ \ 12y_{1}+5y_{2}+12y_{3}\le 34.8,\\ \ \ \ \ \ \ 12y_{1}+12y_{2}+7y_{3}\le 29.1,\\ \ \ \ \ \ \ \ -6y_{1}+y_{2}+y_{3}\le -4.1,\\ \ \ \ \ \ \ 1.0\le y_{1}\le 1.2,\ \ 0.55\le y_{2}\le 0.65,\ \ 1.35\le y_{3}\le 1.45. \end{array} \right. \end{aligned}$$

A.4 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \min \max \left\{ \frac{3y_{1}+y_{2}-2y_{3}+0.8}{2y_{1}-y_{2}+y_{3}},\frac{4y_{1}-2y_{2}+y_{3}}{7y_{1}+3y_{2}-y_{3}},\frac{3y_{1}+2y_{2}-y_{3}+1.9}{y_{1}-y_{2}+y_{3}},\frac{4y_{1}-y_{2}+y_{3}}{8y_{1}+4y_{2}-y_{3}}\right\} \\ {\mathrm{s. t.}}\ \ y_{1}+y_{2}-y_{3}\le 1,\\ \ \ \ \ \ \ -y_{1}+y_{2}-y_{3}\le -1,\\ \ \ \ \ \ \ 12y_{1}+5y_{2}+12y_{3}\le 34.8,\\ \ \ \ \ \ \ 12y_{1}+12y_{2}+7y_{3}\le 29.1,\\ \ \ \ \ \ \ \ -6y_{1}+y_{2}+y_{3}\le -4.1,\\ \ \ \ \ \ \ 1.0\le y_{1}\le 1.2,\ \ 0.55\le y_{2}\le 0.65,\ \ 1.35\le y_{3}\le 1.45.\\ \end{array} \right. \end{aligned}$$

A.5 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \min \max \displaystyle \left\{ \frac{2.1y_{1}+2.2y_{2}-y_{3}+0.8}{1.1y_{1}-y_{2}+1.2y_{3}},\frac{3.1y_{1}-y_{2}+1.3y_{3}}{8.2y_{1}+4.1y_{2}-y_{3}}\right\} \\ {\mathrm{s. t.}}\ \ y_{1}+y_{2}-y_{3}\le 1,\\ \ \ \ \ \ \ -y_{1}+y_{2}-y_{3}\le -1,\\ \ \ \ \ \ \ 12y_{1}+5y_{2}+12y_{3}\le 40,\\ \ \ \ \ \ \ 12y_{1}+12y_{2}+7y_{3}\le 50,\\ \ \ \ \ \ \ \ -6y_{1}+y_{2}+y_{3}\le -2,\\ \ \ \ \ \ \ \ 1.0\le y_{1}\le 1.2, \ \ \ \ 0.55\le y_{2}\le 0.65, \ \ \ \ 1.35\le y_{3}\le 1.45. \end{array} \right. \end{aligned}$$

A.6 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \min \max \left\{ \frac{3y_{1}+4y_{2}-y_{3}+0.5}{2y_{1}-y_{2}+y_{3}+0.5},\frac{3y_{1}-y_{2}+3y_{3}+0.5}{9y_{1}+5y_{2}-y_{3}+0.5},\frac{4y_{1}-y_{2}+5y_{3}+0.5}{11y_{1}+6y_{2}-y_{3}},\frac{5y_{1}-y_{2}+6y_{3}+0.5}{12y_{1}+7y_{2}-y_{3}+0.9}\right\} \\ {\mathrm{s. t.}}\ \ y_{1}+y_{2}-y_{3}\le 1,\\ \ \ \ \ \ \ -y_{1}+y_{2}-y_{3}\le -1,\\ \ \ \ \ \ \ 12y_{1}+5y_{2}+12y_{3}\le 42,\\ \ \ \ \ \ \ 12y_{1}+12y_{2}+7y_{3}\le 55,\\ \ \ \ \ \ \ \ -6y_{1}+y_{2}+y_{3}\le -3,\\ \ \ \ \ \ \ 1.0\le y_{1}\le 2.0, \ \ \ \ 0.50\le y_{2}\le 2.0, \ \ \ \ 0.50\le y_{3}\le 2.0. \end{array} \right. \end{aligned}$$

A.7 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \min \max \left\{ \frac{3y_{1}+4y_{2}-y_{3}+0.9}{2y_{1}-y_{2}+y_{3}+0.5},\frac{3y_{1}-y_{2}+3y_{3}+0.5}{9y_{1}+5y_{2}-y_{3}+0.5},\frac{4y_{1}-y_{2}+5y_{3}+0.5}{11y_{1}+6y_{2}-y_{3}+0.9},\right. \\ \ \ \ \ \ \ \ \ \ \ \ \ \left. \frac{5y_{1}-y_{2}+6y_{3}+0.5}{12y_{1}+7y_{2}-y_{3}+0.9},\frac{6y_{1}-y_{2}+7y_{3}+0.6}{11y_{1}+6y_{2}-y_{3}+0.9}\right\} \\ {\mathrm{s. t.}}\ \ 2y_{1}+y_{2}-y_{3}\le 2,\\ \ \ \ \ \ \ -2y_{1}+y_{2}-2y_{3}\le -1,\\ \ \ \ \ \ \ 11y_{1}+6y_{2}+12y_{3}\le 45,\\ \ \ \ \ \ \ 11y_{1}+13y_{2}+6y_{3}\le 52,\\ \ \ \ \ \ \ \ -7y_{1}+y_{2}+y_{3}\le -2,\\ \ \ \ \ \ \ 1.0\le y_{1}\le 2.0, \ \ \ \ 0.35\le y_{2}\le 0.9, \ \ \ \ 1.0\le y_{3}\le 1.55. \end{array} \right. \end{aligned}$$

A.8 [15, 19].

$$\begin{aligned} \left\{ \begin{array}{ll} \min \max \left\{ \frac{5y_{1}+4y_{2}-y_{3}+0.9}{3y_{1}-y_{2}+2y_{3}+0.5},\frac{3y_{1}-y_{2}+4y_{3}+0.5}{9y_{1}+3y_{2}-y_{3}+0.5},\frac{4y_{1}-y_{2}+6y_{3}+0.5}{12y_{1}+7y_{2}-y_{3}+0.9},\right. \\ \ \ \ \ \ \ \ \ \ \ \ \ \left. \frac{7y_{1}-y_{2}+7y_{3}+0.5}{11y_{1}+9y_{2}-y_{3}+0.9},\frac{7y_{1}-y_{2}+7y_{3}+0.7}{11y_{1}+7y_{2}-y_{3}+0.8}\right\} \\ {\mathrm{s. t.}}\ \ 2y_{1}+2y_{2}-y_{3}\le 3,\\ \ \ \ \ \ \ -2y_{1}+y_{2}-3y_{3}\le -1,\\ \ \ \ \ \ \ 11y_{1}+7y_{2}+12y_{3}\le 47,\\ \ \ \ \ \ \ 13y_{1}+13y_{2}+6y_{3}\le 56,\\ \ \ \ \ \ \ \ -6y_{1}+2y_{2}+3y_{3}\le -1,\\ \ \ \ \ \ \ 1.0\le y_{1}\le 2.0, \ \ \ \ 0.35\le y_{2}\le 0.9, \ \ \ \ 1.0\le y_{3}\le 1.55. \end{array} \right. \end{aligned}$$

A.9.

$$\begin{aligned} \left\{ \begin{array}{ll} \min \ \max \ {\left\{ \frac{\sum \limits _{j=1}^{n} {\hat{d}}_{1j}y_{j}+{\hat{g}}_{1}}{\sum \limits _{j=1}^{n} {\hat{e}}_{1j}y_{j}+{\hat{h}}_{1}},\ldots ,\frac{\sum \limits _{j=1}^{n} {\hat{d}}_{pj}y_{j}+{\hat{g}}_{p}}{\sum \limits _{j=1}^{n} {\hat{e}}_{pj}y_{j}+{\hat{h}}_{p}}\right\} } \\ {\mathrm{s.\ t.}}\ \ \sum \limits _{j=1}^{n} {\hat{a}}_{qj}y_{j}\le {\hat{b}}_{q},\ q=1,2,\ldots ,{\hat{m}},\\ \ \ \ \ \ \ \ y_{j}\ge 0,\ j=1,2,\ldots ,n, \end{array} \right. \end{aligned}$$

where ${\hat{d}}_{ij}, {\hat{e}}_{ij}, {\hat{b}}_{k}$ and ${\hat{a}}_{qj}$ are all randomly produced from [0, 10]; ${\hat{g}}_{i}$ and ${\hat{h}}_{i}$ are all randomly produced from [0, 1].

A.10.

$$\begin{aligned} \left\{ \begin{array}{l} \min \ \max \ \left\{ \frac{\sum \limits _{j=1}^{n}{\widetilde{d}}_{1j}y_{j}+{\widetilde{f}}_{1}}{\sum \limits _{j=1}^{n}{\widetilde{c}}_{1j}y_{j}+{\widetilde{g}}_{1}}, \ldots , \frac{\sum \limits _{j=1}^{n}{\widetilde{d}}_{pj}y_{j}+{\widetilde{f}}_{p}}{\sum \limits _{j=1}^{n}{\widetilde{c}}_{pj}y_{j}+{\widetilde{g}}_{p}}\right\} \\ \mathrm {s.t.}\ \ \ \ \ \ \ \ \ \ \sum \limits _{j=1}^{n}{\widetilde{a}}_{qj}y_{j}\le {\widetilde{b}}_{q},\ \ q=1,2,\ldots ,{\hat{m}},\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y_{j}\ge 0,\ \ j=1,\ldots ,n. \end{array} \right. \end{aligned}$$

where all ${\widetilde{d}}_{ij}$ and ${\widetilde{c}}_{ij}$ are the average number of distributions randomly generated from $[-0.1, 0.1]$, all ${\widetilde{a}}_{qj}$ are the average number of distributions randomly generated from [0.01, 1], all ${\widetilde{b}}_{q}=10$, all constant terms ${\widetilde{f}}_{i}$ and ${\widetilde{g}}_{i}$ of numerators and denominators of ratios satisfy that $\sum \limits _{j=1}^{n}{\widetilde{c}}_{ij}y_{j}+{\widetilde{f}}_{i}>0$ and $\sum \limits _{j=1}^{n}{\widetilde{d}}_{ij}y_{j}+{\widetilde{g}}_{i}>0, i=1,\ldots ,p$.