# Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Solution Methods

**DML**).

**SDM**).

**SDM**).

**CGM**) as suggested in [21]. For the sake of clarity, we describe (

**CGM**) applied to the general optimization problem

**CGM**) Take an arbitrary initial point ${v}^{0}\in V$ and a number $\delta >0$. At the s-th iteration, $s=0,1,\dots $, we have a point ${v}^{s}\in V$ and calculate ${u}^{s}\in V$ as a solution of the linear programming problem

**SDM**). We denote this method as (

**CGDM**). However, this approach requires application of (

**CGM**) many times at each iteration of a single-dimensional optimization algorithm applied to the upper problem (4). At the same time, we can apply the same dual decomposition method to problem (5) and (6). For the sake of simplicity, we rewrite (5) and (6) as follows:

**Algorithm (BS)**. Given an accuracy $\epsilon >0$ and the initial segment $[{p}^{\prime},{p}^{\u2033}]$, we take $\tilde{p}=0.5({p}^{\prime}+{p}^{\prime \prime})$, calculate ${\theta}^{\prime}\left(\tilde{p}\right)$. Then we set ${p}^{\prime}=\tilde{p}$ if ${\theta}^{\prime}\left(\tilde{p}\right)>0$ and ${p}^{\prime}=\tilde{p}$ otherwise, until $({p}^{\u2033}-{p}^{\prime})<\epsilon $.

**Algorithm (SQ)**. Let ${\omega}_{j}={\eta}_{j,1}+\lambda {\phi}_{i,1}$. Define ${J}_{a}=\{j\in J\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}{\omega}_{j}>{p}^{\prime}\}$, set ${y}_{j}^{\ast}=0$ for $j\notin {J}_{a}$ and re-arrange the indices in ${J}_{a}$ to have the descending order for the values of ${\omega}_{j}$. Then find two sequential indices ${j}_{l}$ and ${j}_{l+1}$ in ${J}_{a}$ such that ${\Delta}_{l}<0$ and ${\Delta}_{l+1}>0$, where

## 4. Numerical Experiments

**[Case L]**All the functions ${\mu}_{i}\left({x}_{i}\right)$, ${\phi}_{i}\left({x}_{i}\right)$, and ${\eta}_{j}\left({y}_{j}\right)$ are affine;**[Case QL]**All the functions ${\eta}_{j}\left({y}_{j}\right)$ are affine, all the functions ${\mu}_{i}\left({x}_{i}\right)$ and ${\phi}_{i}\left({x}_{i}\right)$ are quadratic;**[Case EQ]**All the functions $-{\eta}_{j}\left({y}_{j}\right)$ are convex quadratic, all the functions ${\mu}_{i}\left({x}_{i}\right)$ and ${\phi}_{i}\left({x}_{i}\right)$ are convex exponential;**[Case Q]**All the functions $-{\eta}_{j}\left({y}_{j}\right)$, ${\mu}_{i}\left({x}_{i}\right)$, and ${\phi}_{i}\left({x}_{i}\right)$ are convex quadratic;**[Case E]**All the functions $-{\eta}_{j}\left({y}_{j}\right)$, ${\mu}_{i}\left({x}_{i}\right)$, and ${\phi}_{i}\left({x}_{i}\right)$ are convex exponential;**[Case LG]**All the functions $-{\eta}_{j}\left({y}_{j}\right)$, ${\mu}_{i}\left({x}_{i}\right)$, and ${\phi}_{i}\left({x}_{i}\right)$ are convex logarithmic.

- 1.
- Linear functions$$\begin{array}{c}{\eta}_{j}\left({y}_{j}\right)={\eta}_{j,1}{y}_{j}+{\eta}_{j,0},{\eta}_{j,1}>0,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i}\left({x}_{i}\right)={\mu}_{i,1}{x}_{i}+{\mu}_{i,0},{\mu}_{i,1}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i}\left({x}_{i}\right)={\phi}_{i,1}{x}_{i}+{\phi}_{i,0},{\phi}_{i,1}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \end{array}$$$$\begin{array}{c}{\eta}_{j,1}=2\left|\mathrm{sin}(j+1)\right|+1,{\eta}_{j,0}=2\left|\mathrm{sin}\left(2j\right)\right|+1,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i,1}=\left|\mathrm{cos}\left(i\right)\right|+1,{\mu}_{i,0}=2\left|\mathrm{cos}\left(2i\right)\right|+1,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i,1}={\mu}_{i,1},{\phi}_{i,0}={\mu}_{i,0},\phantom{\rule{4pt}{0ex}}i=1,\dots ,m.\hfill \end{array}$$
- 2.
- Quadratic functions$$\begin{array}{c}{\eta}_{j}\left({y}_{j}\right)=0.5{\eta}_{j,2}{y}_{j}^{2}+{\eta}_{j,1}{y}_{j},{\eta}_{j,2}<0,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i}\left({x}_{i}\right)=0.5{\mu}_{i,2}{x}_{i}^{2}+{\mu}_{i,1}{x}_{i},{\mu}_{i,2}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i}\left({x}_{i}\right)=0.5{\phi}_{i,2}{x}_{i}^{2}+{\phi}_{i,1}{x}_{i},{\phi}_{i,2}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \end{array}$$$$\begin{array}{c}{\eta}_{j,2}=-4\left|\mathrm{cos}(2j-1)\right|-4,{\eta}_{j,1}=\left|\mathrm{sin}(j+1)\right|+1,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i,2}=\left|\mathrm{sin}\left(2i\right)\right|+1,{\mu}_{i,1}=\left|\mathrm{cos}\left(i\right)\right|+3,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i,2}={\mu}_{i,2},{\phi}_{i,1}={\mu}_{i,1},\phantom{\rule{4pt}{0ex}}i=1,\dots ,m.\hfill \end{array}$$
- 3.
- Exponential functions$$\begin{array}{c}{\eta}_{j}\left({y}_{j}\right)={\eta}_{j,0}+{\eta}_{j,1}{y}_{j}-{\eta}_{j,2}{e}^{{\eta}_{j,3}{y}_{j}},{\eta}_{j,1},{\eta}_{j,2}>0,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i}\left({x}_{i}\right)={\mu}_{i,0}{e}^{{\mu}_{i,1}{x}_{i}},{\mu}_{i,1},{\mu}_{i,0}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i}\left({x}_{i}\right)={\phi}_{i,0}{e}^{{\phi}_{i,1}{x}_{i}},{\phi}_{i,1},{\phi}_{i,0}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \end{array}$$$$\begin{array}{c}{\eta}_{j,3}=\left|\mathrm{sin}(j+1)\right|+1,{\eta}_{j,2}=2\left|\mathrm{sin}\left(2j\right)\right|+1,\hfill \\ {\eta}_{j,1}=2\left|\mathrm{sin}(j+1)\right|+8,{\eta}_{j,0}=2\left|\mathrm{sin}\left(2j\right)\right|+9,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i,1}=\left|\mathrm{cos}\left(i\right)\right|+1,{\mu}_{i,0}=2\left|\mathrm{cos}\left(2i\right)\right|+1,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i,1}={\mu}_{i,1},{\phi}_{i,0}={\mu}_{i,0},\phantom{\rule{4pt}{0ex}}i=1,\dots ,m.\hfill \end{array}$$
- 4.
- Logarithmic functions$$\begin{array}{c}{\eta}_{j}\left({y}_{j}\right)={\eta}_{j,2}\mathrm{ln}(1+{\eta}_{j,0}+{\eta}_{j,1}{y}_{j}),{\eta}_{j,0},{\eta}_{j,1},{\eta}_{j,2}>0,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i}\left({x}_{i}\right)={\mu}_{i,0}+{\mu}_{i,1}{x}_{i}-\mathrm{ln}(1+{\mu}_{i,2}+{\mu}_{i,3}{x}_{i}),{\mu}_{i,1},{\mu}_{i,2},{\mu}_{i,3}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i}\left({x}_{i}\right)={\phi}_{i,0}+{\phi}_{i,1}{x}_{i}-\mathrm{ln}(1+{\phi}_{i,2}+{\phi}_{i,3}{x}_{i}),{\phi}_{i,1},{\phi}_{i,2},{\phi}_{i,3}>0,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \end{array}$$$$\begin{array}{c}{\eta}_{j,0}=2\left|\mathrm{sin}\left(2j\right)\right|,{\eta}_{j,1}=\left|\mathrm{sin}(j+1)\right|+1,\hfill \\ {\eta}_{j,2}=3\left|\mathrm{sin}\left(2j\right)\right|+1,\phantom{\rule{4pt}{0ex}}j=1,\dots ,J,\hfill \\ {\mu}_{i,0}=2\left|\mathrm{cos}\left(2i\right)\right|+1,{\mu}_{i,1}=\left|\mathrm{cos}\left(i\right)\right|+1,\hfill \\ {\mu}_{i,2}=2\left|\mathrm{cos}\left(2i\right)\right|,{\mu}_{i,3}=\left|\mathrm{cos}\left(i\right)\right|+1,\phantom{\rule{4pt}{0ex}}i=1,\dots ,m,\hfill \\ {\phi}_{i,0}={\mu}_{i,0},{\phi}_{i,1}={\mu}_{i,1},{\phi}_{i,2}={\mu}_{i,2},{\phi}_{i,3}={\mu}_{i,3},\phantom{\rule{4pt}{0ex}}i=1,\dots ,m.\hfill \end{array}$$

**DML**), (

**CGDM**), and (

**BS**), we tested also their modifications adjusted mainly to some particular classes of problems. We applied the method (

**DML**) with adaptive strategy of choosing the inner accuracies and named it (

**DMLA**). In the case where the functions ${\eta}_{j}\left({y}_{j}\right)$ are affine, we applied also the simplified versions of these methods named (

**DMLS**) and (

**DMLAS**), respectively. They solve auxiliary problems (7) and (8) by a simple ordering algorithm in a finite number of iterations and require only one arrangement of buyers’ prices. Methods (

**DML**), (

**DMLA**), (

**DMLS**), (

**DMLAS**) and (

**SDM**) were applied for cases L and QL, where (

**DMLS**) and (

**DMLAS**) showed better performance than (

**DML**) and (

**DMLA**), but (

**SDM**) showed the best results here.

**CGDM**), where (

**CGDM0**) denotes the version with zero initial point for any (

**CGM**), (

**CGDMB**) denotes the version with taking the initial point for any (

**CGM**) in the boundary of the feasible set. We utilized these methods with the inexact line search procedure. Therefore, methods (

**DML**), (

**DMLA**), (

**CGDM0**), (

**CGDMB**), (

**BS**), and (

**SQ**) were applied for Case Q. Here (

**BS**) and (

**SQ**) showed the best performance, and the results of (

**CGDM0**) and (

**CGDMB**) were better than those of (

**DML**) and (

**DMLA**).

**DML**), (

**DMLA**), (

**CGDM0**), (

**CGDMB**), and (

**BS**) were applied for cases EQ, E, and LG. Here (

**BS**) showed the essentially better results than the other methods. Also, (

**DMLA**) showed better performance than (

**DML**), (

**CGDM0**), and (

**CGDMB**) in most test experiments.

**BS**) showed the best results for the nonlinear problems.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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${\mathit{\epsilon}}_{\mathit{\lambda}}$ | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLAS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SDM) |
---|---|---|---|---|---|

${10}^{-1}$ | 0.1680 | 0.0897 | 0.0025 | 0.0024 | 0.0003 |

${10}^{-2}$ | 0.1919 | 0.1159 | 0.0031 | 0.0025 | 0.0004 |

${10}^{-3}$ | 0.2371 | 0.1597 | 0.0047 | 0.0028 | 0.0009 |

${10}^{-4}$ | 0.2728 | 0.2128 | 0.0062 | 0.0046 | 0.0012 |

J | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLAS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SDM) |
---|---|---|---|---|---|

210 | 0.0849 | 0.0489 | 0.0025 | 0.0012 | 0.0004 |

310 | 0.1192 | 0.0712 | 0.0021 | 0.0028 | 0.0004 |

410 | 0.1549 | 0.0928 | 0.0012 | 0.0009 | 0.0005 |

510 | 0.1919 | 0.1159 | 0.0031 | 0.0025 | 0.0005 |

610 | 0.2340 | 0.1368 | 0.0050 | 0.0028 | 0.0005 |

710 | 0.2716 | 0.1590 | 0.0047 | 0.0028 | 0.0003 |

810 | 0.3100 | 0.1834 | 0.0044 | 0.0035 | 0.0004 |

910 | 0.3487 | 0.2056 | 0.0050 | 0.0062 | 0.0009 |

1010 | 0.3872 | 0.2287 | 0.0072 | 0.0042 | 0.0010 |

m | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLAS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SDM) |
---|---|---|---|---|---|

3 | 0.1781 | 0.0988 | 0.0024 | 0.0021 | 0.0004 |

9 | 0.1966 | 0.1130 | 0.0019 | 0.0012 | 0.0004 |

15 | 0.1976 | 0.1156 | 0.0018 | 0.0009 | 0.0001 |

21 | 0.2004 | 0.1157 | 0.0018 | 0.0018 | 0.0003 |

27 | 0.1953 | 0.1140 | 0.0027 | 0.0012 | 0.0007 |

33 | 0.1958 | 0.1153 | 0.0028 | 0.0021 | 0.0007 |

39 | 0.1994 | 0.1174 | 0.0024 | 0.0022 | 0.0004 |

45 | 0.2010 | 0.1175 | 0.0040 | 0.0022 | 0.0005 |

${\mathit{\epsilon}}_{\mathit{\lambda}}$ | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLAS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SDM) |
---|---|---|---|---|---|

${10}^{-1}$ | 0.1665 | 0.0906 | 0.0025 | 0.0021 | 0.0003 |

${10}^{-2}$ | 0.1959 | 0.1133 | 0.0028 | 0.0024 | 0.0004 |

${10}^{-3}$ | 0.2346 | 0.1603 | 0.0049 | 0.0028 | 0.0006 |

${10}^{-4}$ | 0.2762 | 0.2144 | 0.0053 | 0.0041 | 0.0007 |

J | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLAS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SDM) |
---|---|---|---|---|---|

210 | 0.0814 | 0.0478 | 0.0022 | 0.0006 | 0.0001 |

310 | 0.1155 | 0.0703 | 0.0025 | 0.0012 | 0.0003 |

410 | 0.1593 | 0.0918 | 0.0015 | 0.0027 | 0.0002 |

510 | 0.1959 | 0.1133 | 0.0028 | 0.0024 | 0.0003 |

610 | 0.2331 | 0.1379 | 0.0031 | 0.0021 | 0.0004 |

710 | 0.2736 | 0.1598 | 0.0040 | 0.0015 | 0.0005 |

810 | 0.3115 | 0.1855 | 0.0055 | 0.0031 | 0.0004 |

910 | 0.3522 | 0.2075 | 0.0044 | 0.0019 | 0.0004 |

1010 | 0.3904 | 0.2305 | 0.0050 | 0.0028 | 0.0006 |

m | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLAS) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SDM) |
---|---|---|---|---|---|

3 | 0.1753 | 0.0984 | 0.0056 | 0.0009 | 0.0001 |

9 | 0.1953 | 0.1153 | 0.0031 | 0.0016 | 0.0003 |

15 | 0.1981 | 0.1187 | 0.0031 | 0.0019 | 0.0002 |

21 | 0.1965 | 0.1196 | 0.0021 | 0.0009 | 0.0002 |

27 | 0.1924 | 0.1141 | 0.0025 | 0.0019 | 0.0001 |

33 | 0.1958 | 0.1163 | 0.0034 | 0.0025 | 0.0003 |

39 | 0.1977 | 0.1162 | 0.0041 | 0.0015 | 0.0001 |

45 | 0.1964 | 0.1158 | 0.0031 | 0.0015 | 0.0006 |

${\mathit{\epsilon}}_{\mathit{\lambda}}$ | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SQ) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|---|

${10}^{-1}$ | 0.2815 | 0.1488 | 0.0283 | 0.0762 | 0.0012 | 0.0018 |

${10}^{-2}$ | 0.3271 | 0.1929 | 0.0474 | 0.1116 | 0.0015 | 0.0019 |

${10}^{-3}$ | 0.3909 | 0.2669 | 0.1049 | 0.1919 | 0.0026 | 0.0040 |

${10}^{-4}$ | 0.4575 | 0.3570 | 0.1486 | 0.2534 | 0.0028 | 0.0058 |

J | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SQ) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|---|

210 | 0.1377 | 0.0833 | 0.0113 | 0.0367 | 0.0006 | 0.0009 |

310 | 0.2004 | 0.1179 | 0.0208 | 0.0580 | 0.0012 | 0.0019 |

410 | 0.2657 | 0.1564 | 0.0321 | 0.0828 | 0.0009 | 0.0018 |

510 | 0.3271 | 0.1929 | 0.0474 | 0.1116 | 0.0015 | 0.0019 |

610 | 0.3900 | 0.2302 | 0.0641 | 0.1396 | 0.0027 | 0.0030 |

710 | 0.4540 | 0.2682 | 0.0728 | 0.1568 | 0.0019 | 0.0034 |

810 | 0.5181 | 0.3052 | 0.0862 | 0.1804 | 0.0022 | 0.0028 |

910 | 0.5809 | 0.3437 | 0.0934 | 0.1960 | 0.0028 | 0.0043 |

1010 | 0.6434 | 0.3793 | 0.1021 | 0.2110 | 0.0025 | 0.0046 |

m | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (SQ) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|---|

3 | 0.3028 | 0.1663 | 0.0465 | 0.0937 | 0.0003 | 0.0031 |

9 | 0.3203 | 0.1881 | 0.0546 | 0.0986 | 0.0015 | 0.0021 |

15 | 0.3250 | 0.1956 | 0.0668 | 0.1205 | 0.0012 | 0.0019 |

21 | 0.3278 | 0.1952 | 0.0541 | 0.1131 | 0.0015 | 0.0022 |

27 | 0.3271 | 0.1957 | 0.0462 | 0.1075 | 0.0012 | 0.0028 |

33 | 0.3300 | 0.1946 | 0.0462 | 0.1111 | 0.0006 | 0.0034 |

39 | 0.3353 | 0.1972 | 0.0365 | 0.1037 | 0.0003 | 0.0028 |

45 | 0.3337 | 0.1974 | 0.0333 | 0.0968 | 0.0006 | 0.0021 |

${\mathit{\epsilon}}_{\mathit{\lambda}}$ | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

${10}^{-1}$ | 0.2781 | 0.1478 | 0.0833 | 0.3225 | 0.0040 |

${10}^{-2}$ | 0.3281 | 0.1938 | 0.1497 | 0.4153 | 0.0043 |

${10}^{-3}$ | 0.3913 | 0.2666 | 0.2547 | 0.6068 | 0.0088 |

${10}^{-4}$ | 0.4534 | 0.3572 | 0.3596 | 0.7553 | 0.0103 |

J | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

210 | 0.1382 | 0.0845 | 0.0293 | 0.0824 | 0.0030 |

310 | 0.2028 | 0.1187 | 0.0674 | 0.1753 | 0.0018 |

410 | 0.2644 | 0.1555 | 0.1127 | 0.2857 | 0.0021 |

510 | 0.3281 | 0.1938 | 0.1497 | 0.4153 | 0.0043 |

610 | 0.3906 | 0.2294 | 0.1693 | 0.5499 | 0.0053 |

710 | 0.4510 | 0.2669 | 0.2031 | 0.6742 | 0.0050 |

810 | 0.5161 | 0.3075 | 0.2312 | 0.7656 | 0.0059 |

910 | 0.5766 | 0.3449 | 0.2718 | 0.8646 | 0.0097 |

1010 | 0.6421 | 0.3819 | 0.3184 | 0.9579 | 0.0085 |

m | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

3 | 0.3035 | 0.1671 | 0.0762 | 0.3703 | 0.0040 |

9 | 0.3188 | 0.1871 | 0.0746 | 0.3753 | 0.0030 |

15 | 0.3231 | 0.1932 | 0.1228 | 0.4345 | 0.0046 |

21 | 0.3278 | 0.1949 | 0.1459 | 0.4269 | 0.0053 |

27 | 0.3278 | 0.1924 | 0.1193 | 0.3479 | 0.0047 |

33 | 0.3309 | 0.1955 | 0.1215 | 0.3411 | 0.0041 |

39 | 0.3303 | 0.1981 | 0.1124 | 0.2996 | 0.0053 |

45 | 0.3325 | 0.1949 | 0.1166 | 0.2928 | 0.0052 |

${\mathit{\epsilon}}_{\mathit{\lambda}}$ | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

${10}^{-1}$ | 0.2787 | 0.1468 | 1.9862 | 2.3146 | 0.0071 |

${10}^{-2}$ | 0.3190 | 0.1884 | 3.2766 | 3.6343 | 0.0083 |

${10}^{-3}$ | 0.3887 | 0.2642 | 5.1906 | 5.6561 | 0.0146 |

${10}^{-4}$ | 0.4446 | 0.3522 | 6.9235 | 7.4660 | 0.0175 |

J | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

210 | 0.1334 | 0.0797 | 0.9962 | 1.0643 | 0.0028 |

310 | 0.1966 | 0.1143 | 1.6407 | 1.7865 | 0.0078 |

410 | 0.2559 | 0.1506 | 2.3843 | 2.6225 | 0.0081 |

510 | 0.3190 | 0.1884 | 3.2766 | 3.6343 | 0.0083 |

610 | 0.3793 | 0.2231 | 4.1481 | 4.6926 | 0.0090 |

710 | 0.4399 | 0.2612 | 4.7769 | 5.4312 | 0.0136 |

810 | 0.5057 | 0.2981 | 5.6989 | 6.5140 | 0.0140 |

910 | 0.5659 | 0.3353 | 6.2093 | 7.1038 | 0.0156 |

1010 | 0.6309 | 0.3715 | 7.0919 | 8.0395 | 0.0168 |

m | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

3 | 0.2981 | 0.1662 | 2.4191 | 2.8496 | 0.0072 |

9 | 0.3128 | 0.1828 | 2.8231 | 3.2803 | 0.0068 |

15 | 0.3203 | 0.1896 | 3.6925 | 4.1387 | 0.0106 |

21 | 0.3216 | 0.1918 | 3.6565 | 4.0279 | 0.0103 |

27 | 0.3225 | 0.1881 | 3.2027 | 3.5053 | 0.0089 |

33 | 0.3247 | 0.1907 | 1.7065 | 1.9672 | 0.0109 |

39 | 0.3278 | 0.1916 | 2.9506 | 3.2169 | 0.0099 |

45 | 0.3264 | 0.1906 | 2.9452 | 3.1939 | 0.0113 |

${\mathit{\epsilon}}_{\mathit{\lambda}}$ | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

${10}^{-1}$ | 0.3617 | 0.1921 | 0.1438 | 0.2213 | 0.0028 |

${10}^{-2}$ | 0.4240 | 0.2528 | 0.2318 | 0.3150 | 0.0046 |

${10}^{-3}$ | 0.5090 | 0.3452 | 0.3934 | 0.5133 | 0.0051 |

${10}^{-4}$ | 0.5900 | 0.4627 | 0.5259 | 0.6784 | 0.0053 |

J | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

210 | 0.1780 | 0.1054 | 0.0427 | 0.0703 | 0.0016 |

310 | 0.2585 | 0.1533 | 0.0811 | 0.1240 | 0.0019 |

410 | 0.3391 | 0.2009 | 0.1565 | 0.2247 | 0.0034 |

510 | 0.4240 | 0.2528 | 0.2318 | 0.3150 | 0.0046 |

610 | 0.5109 | 0.3013 | 0.3136 | 0.4281 | 0.0041 |

710 | 0.5962 | 0.3527 | 0.4190 | 0.5507 | 0.0043 |

810 | 0.6819 | 0.4019 | 0.5087 | 0.6494 | 0.0046 |

910 | 0.7675 | 0.4546 | 0.6041 | 0.7487 | 0.0046 |

1010 | 0.8540 | 0.5038 | 0.7293 | 0.8880 | 0.0064 |

m | ${\mathit{T}}_{\mathit{\epsilon}}$: (DML) | ${\mathit{T}}_{\mathit{\epsilon}}$: (DMLA) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDM0) | ${\mathit{T}}_{\mathit{\epsilon}}$: (CGDMB) | ${\mathit{T}}_{\mathit{\epsilon}}$: (BS) |
---|---|---|---|---|---|

3 | 0.3975 | 0.2203 | 0.6333 | 0.5357 | 0.0012 |

9 | 0.4262 | 0.2493 | 0.4346 | 0.4997 | 0.0015 |

15 | 0.4302 | 0.2544 | 0.3605 | 0.4422 | 0.0027 |

21 | 0.4303 | 0.2569 | 0.2607 | 0.3452 | 0.0031 |

27 | 0.4234 | 0.2494 | 0.1863 | 0.2740 | 0.0019 |

33 | 0.4246 | 0.2522 | 0.1391 | 0.1968 | 0.0040 |

39 | 0.4253 | 0.2528 | 0.1515 | 0.2444 | 0.0021 |

45 | 0.4278 | 0.2527 | 0.1375 | 0.2346 | 0.0040 |

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## Share and Cite

**MDPI and ACS Style**

Konnov, I.; Kashuba, A.; Laitinen, E. Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users. *Math. Comput. Appl.* **2018**, *23*, 31.
https://doi.org/10.3390/mca23020031

**AMA Style**

Konnov I, Kashuba A, Laitinen E. Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users. *Mathematical and Computational Applications*. 2018; 23(2):31.
https://doi.org/10.3390/mca23020031

**Chicago/Turabian Style**

Konnov, Igor, Aleksey Kashuba, and Erkki Laitinen. 2018. "Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users" *Mathematical and Computational Applications* 23, no. 2: 31.
https://doi.org/10.3390/mca23020031