# Large Scale Global Optimization Algorithms for IoT Networks: A Comparative
Study

Sotirios K. Goudos1, Achilles D. Boursianis1, Ali Wagdy Mohamed2, Shaohua
Wan3, Panagiotis Sarigiannidis4,
George K. Karagiannidis5, and Ponnuthurai N. Suganthan6  1 ELEDIA@AUTH,School
of Physics,Aristotle University of Thessaloniki,Thessaloniki, Greece
2Operations Research Department, Faculty of Graduate Studies for Statistical
Research,Cairo University, Giza, Egypt,
School of Engineering and Applied Sciences, Wireless Intelligent Networks
Center (WINC),
Nile University, Giza, Egypt  3School of Information and Safety Engineering,
Zhongnan University of Economics and Law, Wuhan, China  4Department of
Informatics and Telecommunications Engineering, University of Western
Macedonia, Kozani, Greece  5School of Electrical and Computer Engineering,
Aristotle University of Thessaloniki,Thessaloniki, Greece  6School of
Electrical Electronic Engineering,Nanyang Technological University, Singapore

###### Abstract

The advent of Internet of Things (IoT) has bring a new era in communication
technology by expanding the current inter-networking services and enabling the
machine-to-machine communication. IoT massive deployments will create the
problem of optimal power allocation. The objective of the optimization problem
is to obtain a feasible solution that minimizes the total power consumption of
the WSN, when the error probability at the fusion center meets certain
criteria. This work studies the optimization of a wireless sensor network
(WNS) at higher dimensions by focusing to the power allocation of
decentralized detection. More specifically, we apply and compare four
algorithms designed to tackle Large scale global optimization (LGSO) problems.
These are the memetic linear population size reduction and semi-parameter
adaptation (MLSHADE-SPA), the contribution-based cooperative coevolution
recursive differential grouping (CBCC-RDG3), the differential grouping with
spectral clustering-differential evolution cooperative coevolution (DGSC-
DECC), and the enhanced adaptive differential evolution (EADE). To the best of
the authors knowledge, this is the first time that LGSO algorithms are applied
to the optimal power allocation problem in IoT networks. We evaluate the
algorithms performance in several different cases by applying them in cases
with 300, 600 and 800 dimensions.

###### Index Terms:

Internet of Things, optimal power allocation, wireless sensor network, large
scale global optimization

##  I Introduction

Large scale global optimization (LGSO) problems are attracting significant
attention by the researchers over the last years. Several optimization
problems in wireless communications can be inherently extended to higher
dimensions. The Internet of Things (IoT) paradigm, as well as the advent of 5G
cellular communication systems, will use hundreds of wireless sensor nodes in
a limited space that require an optimal power allocation scheme. These sets of
wireless sensors work together to monitor an area, gather data from specific
applications, and constitute a wireless sensor network (WSN). There is a huge
number of applications that WSNs are utilized. Among the most representative
examples are included the monitoring of environmental (both indoors/outdoors)
parameters, the monitoring of the power grid, the tracking and targeting of
military targets, the seismic waves sensing, and the healthcare and human
activity monitoring [1, 2, 3].

In wireless sensor networks, if the detection mechanism is based on a
decentralized scheme, each sensor node, after a local pre-processing of its
own observations, transmits a summary of the pre-processed data to the _fusion
center_ , i.e. the central node of the wireless sensor architecture. Then, the
central node applies a final decision based on a set of few hypotheses. Taking
into consideration the given description of the decentralized scheme, the
fusion center does not have direct access to the initial un-processed
observation data. Thus, the data rate requirements of the decentralized scheme
are significantly lower than in a centralized scheme (where all the nodes of
the WSN send all the information to the fusion center directly). This is
because only a few bits of the observation data are transmitted [4]. This
aforementioned problem, i.e the distributed detection and fusion under
specific constraints in WSNs, has attracted a lot of attention in the
literature [5, 6, 7].

In this paper, we assume that the fusion center of a WSN is decentralized and
operates on a binary hypothesis testing problem. Also, the observations of the
wireless sensor nodes are correlated. In this context, we can define the goal
of the optimization problem as the minimization of the total power, under the
application of specific criteria to the error probability detection of the
central node, having the optimal allocation of the total power resources.

The application of EAs to this problem has already been discussed in the
literature by utilizing the Particle Swarm Optimization (PSO) [8], the
Differential Evolution (DE)[9], a hybrid Biogeography Based Optimization -
Differential Evolution (BBO-DE) algorithm [10], and a recently introduced
hybrid TLBO-Jaya algorithm [11].

However, to the best of the authors knowledge, this is the first time that
LGSO problem-oriented algorithms are applied to the previously described
optimal power allocation problem. The novelty in our work lies also on the
fact that we address for the first time the optimal power allocation problem
in higher dimensions.

In detail, we apply four state-of-art EAs, namely the memetic linear
population size reduction and semi-parameter adaptation (MLSHADE-SPA) [12],
the contribution-based cooperative coevolution recursive differential grouping
(CBCC-RDG3) [13], the differential grouping with spectral clustering-
differential evolution cooperative coevolution (DGSC-DECC) [14], and the
enhanced adaptive differential evolution (EADE) [15]. We evaluate the above
algorithms on the optimal power allocation problem in IoT networks with
different settings and various dimensions.

##  II Problem Description

We assume that a WSN, in which a decentralized detection scheme is utilized,
has a fusion center and $L$ sensor nodes that are spatially distributed. The
observed signal vector can be expressed as
$\mathbf{x}={{[{{x}_{1}},\,{{x}_{2}},...,{{x}_{L}}]}^{T}}$. Also, we assume
that within the decentralized scheme two local observation states can occur,
i.e. the absence or the presence of the signal. Thus, each node checks the
binary hypothesis problem, by examining the hypotheses ${{H}_{0}}$ (absence of
signal) and ${{H}_{1}}$ (presences of signal). The initial probabilities of
the hypotheses are defined as $P\left({{H}_{0}}\right)={{\pi}_{0}}$ and
$P\left({{H}_{1}}\right)={{\pi}_{1}}$.

If we consider the constant signal detection problem, in which an additive
Gaussian noise is applied, and having the assumption that the local
observation ${z}_{k}$ is achieved at the $k$-th node, the given problem can be
formulated as [10], [8]

$\displaystyle\begin{array}[]{l}{H_{0}}\,:\,{{z}_{k}}={{v}_{k}},\,\,\,k=1,2,...,L\\\
{H_{1}}\,:\,{{z}_{k}}={{x}_{k}}+{{v}_{k}},\,\,\,k=1,2,...,L\end{array}$ (3)

Moreover, we assume that the additive local noise, ${{v}_{k}}$, is Gaussian
distributed noise with mean value equal to zero and variance expressed as
$\sigma_{v}^{2}$. The signal ${{x}_{k}}$ is a known positive constant signal,
so that ${{x}_{k}}=m,\,\,k=1,2,..,L$, for all sensors. The local observation
of signal-to-noise ratio (SNR) is given by $\gamma={{m}^{2}}/\sigma_{v}^{2}$.
In a vector form, the observation of the SNR is expressed as

$\mathbf{z}=\mathbf{x}+\mathbf{v}$ (4)

where $\mathbf{v}={{[{{v}_{1}},\,{{v}_{2}},...,{{v}_{L}}]}^{T}}$ is a Gaussian
vector of a zero mean value and a covariance matrix
${\boldsymbol{\Sigma}}_{\mathbf{v}}$. Each sensor node applies a local
decision ${{u}_{k}({z}_{k})}$. Additionally, an amplified version of the
observation signal is re-transmitted to the fusion center by each node of the
WSN [10] and is given by

${{u}_{k}({z}_{k})}={g}_{k}{z}_{k},\,\,k=1,2,..,L$ (5)

where ${g}_{k}$ denotes the gain at the $k$-th node that is subject to
attenuation and fading. Therefore, the received signal ${r}_{k}$, which is
originated from the $k$-th node, arrives at the fusion center under the two
previously mentioned hypotheses [10, 8]. They are expressed as

$\displaystyle\begin{array}[]{l}{H_{0}}\,:\,{{r}_{k}}=\,\,{{n}_{k}};\,\,\,k=1,2,...,L\\\
{H_{1}}\,:\,{{r}_{k}}=\,\,{{h}_{k}}{{g}_{k}}{{x}_{k}}+{{n}_{k}};\,\,\,k=1,2,...,L\end{array}$
(8)

where

  * •

${{h}_{k}}$ denotes the channel fading coefficient,

  * •

${{g}_{k}}$ denotes the gain,

  * •

${{n}_{k}}={{h}_{k}}{{g}_{k}}{{v}_{k}}+{{w}_{k}}$ is the effective noise at
the fusion center having zero mean value, and

  * •

${{w}_{k}}$ is the receiver noise that can be analyzed to a sequence of
independent and identically distributed components having a zero mean value
and a variance equal to $\sigma_{w}^{2}$.

Therefore, the covariance matrix of the effective noise vector ${\mathbf{n}}$
can be expressed as

${\boldsymbol{\Sigma}}_{\mathbf{n}}={{h}_{k}}{{g}_{k}}{\boldsymbol{\Sigma}}_{\mathbf{v}}{{g}_{k}}{{h}_{k}}+{\boldsymbol{\Sigma}}_{\mathbf{w}}$
(9)

where
${\boldsymbol{\Sigma}}_{\mathbf{w}}={\sigma}_{\mathbf{w}}^{\mathbf{2}}\mathbf{I}$
is the covariance matrix of the receiver noise and $\mathbf{I}$ is the
$L\times L$ identity matrix. In most of the cases, the observations that are
recorded by the sensors are correlated. If we assume that between adjacent
nodes the space is equally distributed in a straight line having a distance of
$d$, the correlation between noise samples originated by nodes $i$ and $j$ is
${{\rho}^{d\left|i-j\right|}}$, with $\left|\rho\right|\leq 1$. In 9,
${{\boldsymbol{\Sigma}}_{\mathbf{v}}}$ can be expressed using a symmetric
Hermitian Toeplitz matrix [10, 8] in the following form

${{\Sigma}_{v}}=\sigma_{v}^{2}\left[\begin{matrix}1&\rho^{d}&.\,\,.\,\,.&{{\rho}^{d(L-2)}}&{{\rho}^{d(L-1)}}\\\
\rho^{d}&1&.\,\,.\,\,.&{{\rho}^{d(L-3)}}&{{\rho}^{d(L-2)}}\\\
.&.&.\,\,.\,\,.&.&.\\\
{{\rho}^{d(L-1)}}&{{\rho}^{d(L-2)}}&.\,\,.\,\,.&\rho^{d}&1\\\
\end{matrix}\right]$ (10)

It is written in vector notation as

$\mathbf{r=Ax+n}$ (11)

where

  * •

$\mathbf{r}={{[{{r}_{1}},\,{{r}_{2}},...,{{r}_{L}}]}^{T}}$ denotes the
received information vector,

  * •

$\mathbf{n}={{[{{n}_{1}},\,{{n}_{2}},...,{{n}_{L}}]}^{T}}$ denotes the noise
vector, and

  * •

matrix $\mathbf{A}$ is given by
$\mathbf{A}=diag\left({{h}_{1}}{{g}_{1}},{{h}_{2}}{{g}_{2}},...,{{h}_{L}}{{g}_{L}}\right)$.

The observations r at the fusion center are expressed as [10, 8]

$\displaystyle\begin{array}[]{l}{H_{0}}:{\bf{r}}\sim
N\left({0,{{\boldsymbol{\Sigma}}_{\bf{n}}}}\right)\\\ {H_{1}}:{\bf{r}}\sim
N\left({{\bf{Am,}}{{\boldsymbol{\Sigma}}_{\bf{n}}}}\right)\end{array}$ (14)

Moreover, the covariance matrix of the noise at the fusion center is given by
[10, 8]

${{\boldsymbol{\Sigma}}_{\mathbf{n}}}\mathbf{=A}{{\boldsymbol{\Sigma}}_{\mathbf{v}}}\boldsymbol{A+\sigma}_{\mathbf{w}}^{\mathbf{2}}\mathbf{I}$
(15)

where $\mathbf{m}$ is the $L$-length vector with all components equal to $m$.
Taking into consideration the threshold $\tau={{\pi}_{0}}/{{\pi}_{1}}$, and
assuming the minimum probability of Bayesian fusion error, the optimum
Bayesian decision rule is mathematically formulated as [8]

$\displaystyle\delta({\bf{r}})=\left\\{\begin{array}[]{l}1\,\,\,\,if\,T({\bf{r}})\geq\ln\tau\\\
0\,\,\,if\,T({\bf{r}})<\ln\tau\end{array}\right.$ (18)

The optimal procedure to apply the decision between the two hypotheses denotes
the threshold rule on a log-likelihood ratio (LLR) of the observation vector.
Thus, the LLR for the detection problem can be written as [10]

$T\left({\bf{r}}\right)=m{{\bf{e}}^{\bf{T}}}{\boldsymbol{f}{A\Sigma}}_{\bf{n}}^{{\bf{-1}}}{\bf{r}}-\frac{1}{2}{m^{2}}{{\bf{e}}^{\bf{T}}}{\boldsymbol{A\Sigma}}_{\bf{n}}^{{\bf{-1}}}{\bf{Ae}}$
(19)

For the given detection problem, the LLR distribution under the two hypotheses
is given by [10, 8]

$\displaystyle{H_{0}}:T\left({\bf{r}}\right)\sim
N\left({-\frac{1}{2}{m^{2}}{{\bf{e}}^{\bf{T}}}{\boldsymbol{A\Sigma}}_{\bf{n}}^{{\bf{-1}}}{\bf{Ae}},{m^{2}}{{\bf{e}}^{\bf{T}}}{\boldsymbol{A\Sigma}}_{\bf{n}}^{{\bf{-1}}}{\bf{Ae}}}\right)$
(20) $\displaystyle{H_{1}}:T\left({\bf{r}}\right)\sim
N\left({\frac{1}{2}{m^{2}}{{\bf{e}}^{\bf{T}}}{\boldsymbol{A\Sigma}}_{\bf{n}}^{{\bf{-1}}}{\bf{Ae}},{m^{2}}{{\bf{e}}^{\bf{T}}}{\boldsymbol{A\Sigma}}_{\bf{n}}^{{\bf{-1}}}{\bf{Ae}}}\right)$
(21)

Based on [10, 8, 11], if we further assume that the two hypotheses are equally
probable, then the threshold is $\tau=1$. The fusion error probability, i.e.
the fusion center selects ${{H}_{1}}$ when ${{H}_{0}}$ is true, or ${{H}_{0}}$
when ${{H}_{1}}$ is true, is expressed as

${{P}_{e}}=Q\left(\frac{1}{2}\sqrt{{{m}^{2}}{{\mathbf{e}}^{\mathbf{T}}}\boldsymbol{A\Sigma}_{\mathbf{n}}^{\mathbf{-1}}\mathbf{Ae}}\right)$
(22)

where $Q\left(.\right)$ denotes the Gaussian Q-function. Therefore, the
optimal power allocation problem is to obtain a set of $L$ optimal sensor gain
values $\mathbf{g}=({{g}_{1}},{{g}_{2}},...,{{g}_{L}})$, which minimize the
total power, by keeping the fusion error probability smaller than a specified
criterion $\varepsilon$, [10]. The mathematical formulation of the
optimization problem can be expressed as

$\displaystyle\begin{array}[]{l}\min\,\,\,\,f\left({\bf{g}}\right)=\sum\nolimits_{\ell=1}^{L}{g_{\ell}^{2}}\\\
{}\,{}\,\,\,\,\zeta\left({\bf{g}}\right)=Q\left({\frac{1}{2}\sqrt{{m^{2}}{{\bf{e}}^{\bf{T}}}{\boldsymbol{A\Sigma}}_{\bf{n}}^{{\bf{-1}}}{\bf{Ae}}}}\right)-\varepsilon\leq
0\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\psi_{l}}\left({\bf{g}}\right)=-{g_{\ell}}\leq
0,\,\,l=1,2,...,L\end{array}$ (26)

If we apply a penalty function to the above optimization problem, we can
combine the objective and constraint functions to a single objective function.
As in [11], we utilize a dynamically modified penalty function, because of its
effectiveness against a static penalty function approach [16]. As a result,
the objective function is formulated as

$\displaystyle\begin{array}[]{l}F\left({\bf{g}}\right)=f\left({\bf{g}}\right)+It\left[{\sum\limits_{i=0}^{L}{\theta\left({{q_{i}}\left({\bf{g}}\right)}\right){q_{i}}{{\left({\bf{g}}\right)}^{\lambda\left({{q_{i}}\left({\bf{g}}\right)}\right)}}}}\right]\\\
{q_{i}}\left({\bf{g}}\right)=\left\\{\begin{array}[]{l}\max\left\\{{0,\zeta\left({\bf{g}}\right)}\right\\},\,\,\,\mathrm{if}\,i=0\\\
\max\left\\{{0,{\psi_{i}}\left({\bf{g}}\right)}\right\\},\,\,\mathrm{otherwise}\end{array}\right.\\\
\lambda\left(x\right)=\left\\{\begin{array}[]{l}1\,\,\,\,if\,x<1\\\
2\,\,\,otherwise\end{array}\right.\\\
\theta\left(x\right)=\left\\{\begin{array}[]{l}10\,\,\,\,\,if\,x\leq 0.1\\\
100\,\,\,\,\,\,if\,\,\,0.1<x\leq 1\\\
300\,\,\,otherwise\end{array}\right.\end{array}$ (38)

where

  * •

$It$ denotes the current iteration number of the algorithm,

  * •

$\theta\left({{q}_{i}}\left(\mathbf{g}\right)\right)$ is a multi-stage
assignment function, and

  * •

$\lambda\left({{q}_{i}}\left(\mathbf{g}\right)\right)$ represents the power of
the penalty function.

TABLE I: Algorithms average results for $L=300$ sensors, $\gamma=10dB$. The
smaller values are in bold font. D=300
---
$\rho$ | $\varepsilon$ | MLSHADE-SPA | DGSC-DECC | CBCC-RDG3 |  EADE
0 | 0.1 | 1.785 | 208.517 | 19.600 | 680.660
0.05 | 3.078 | 213.802 | 1.754 | 734.090
0.01 | 7.966 | 238.207 | 4.246 | 713.802
0.001 | 15.750 | 206.634 | 8.862 | 718.025
0.01 | 0.1 | 1.709 | 200.217 | 19.583 | 752.375
0.05 | 3.240 | 208.556 | 1.789 | 741.533
0.01 | 7.950 | 221.228 | 4.260 | 761.676
0.001 | 14.745 | 212.173 | 8.922 | 697.972
0.1 | 0.1 | 2.105 | 186.664 | 19.576 | 718.065
0.05 | 3.436 | 215.164 | 1.818 | 719.622
0.01 | 8.157 | 206.904 | 4.454 | 689.980
0.001 | 16.081 | 243.721 | 9.395 | 729.020
0.5 | 0.1 | 2.372 | 195.501 | 21.634 | 726.920
0.05 | 3.568 | 242.939 | 2.125 | 862.677
0.01 | 8.267 | 228.090 | 5.595 | 729.278
0.001 | 17.143 | 212.832 | 12.701 | 702.204
TABLE II: Algorithms average results for $L=600,800$ sensors, $\rho=0$, $\gamma=10dB$. The smaller values are in bold font. $D$ | $\varepsilon$ | MLSHADE-SPA | DGSC-DECC | CBCC-RDG3 |  EADE
---|---|---|---|---|---
600 | 0.1 | 2.727 | 11937.294 | 573.896 | 12344.849
0.05 | 4.789 | 11964.444 | 1.802 | 12447.193
0.01 | 9.971 | 11944.247 | 3.838 | 12420.642
0.001 | 18.762 | 12012.850 | 7.530 | 12385.260
800 | 0.1 | 3.623 | 7789.121 | 45000.000 | 9943.722
0.05 | 5.432 | 8088.043 | 2.189 | 10126.987
0.01 | 11.164 | 8043.257 | 4.594 | 10845.345
0.001 | 20.813 | 7399.785 | 8.789 | 10041.129
TABLE III: Average Rankings achieved by Friedman test. Algorithm | Average Rank | Normalized values | Rank
---|---|---|---
MLSHADE-SPA | 1.75 | 1.31 | 2
DGSC-DECC | 2.96 | 2.22 | 3
CBCC-RDG3 | 1.33 | 1.00 | 1
EADE | 3.96 | 2.97 | 4
TABLE IV: Wilcoxon signed-rank test between CBCC-RDG3 and the other algorithms.The bold font indicates values below significance level. CBCC-RDG3 vs | p-value
---|---
MLSHADE-SPA | 3.426E-02
DGSC-DECC | 4.000E-05
EADE | 7.000E-06

(a)

(b)

(c)

(d)

Figure 1: Convergence rate graph for
$L=300,\gamma=10\,\mathrm{dB},\rho=0.1,a)\varepsilon=0.001,b)\varepsilon=0.01,c)\varepsilon=0.05,d)\varepsilon=0.1$.

(a)

(b)

(c)

(d)

Figure 2: Convergence rate graph for
$L=300,\gamma=10\,\mathrm{dB},\rho=0.5,a)\varepsilon=0.001,b)\varepsilon=0.01,c)\varepsilon=0.05,d)\varepsilon=0.1$.

(a)

(b)

(c)

(d)

Figure 3: Convergence rate graph for
$L=600,\gamma=10\,\mathrm{dB},\rho=0,a)\varepsilon=0.001,b)\varepsilon=0.01,c)\varepsilon=0.05,d)\varepsilon=0.1$.

(a)

(b)

(c)

(d)

Figure 4: Convergence rate graph for
$L=800,\gamma=10\,\mathrm{dB},\rho=0,a)\varepsilon=0.001,b)\varepsilon=0.01,c)\varepsilon=0.05,d)\varepsilon=0.1$.

##  III Large Scale Global Optimization algorithms

The algorithms that we will apply for the optimal power allocation problem can
be classified into two categories; those that solve the problem directly,
either by using several algorithms and their hybridization, and those that use
a decomposition strategy, as well as a variable grouping strategy. The first
category includes the MLSHADE-SPA [12] and EADE [15] algorithms, whereas CBCC-
RDG3 [13] and DGSC-DECC [14] algorithms fall into the second category.

###  III-A Enhanced adaptive differential evolution

Mohamed [15] proposed an enhanced adaptive differential evolution (EADE)
algorithm, which is a non-Cooperative Co-evolution method, to provide feasible
solutions in large scale optimization problems. EADE includes two novel
contributions. Firstly, in order to benefit from the available information of
the whole population, a novel mutation scheme has been introduced. The
proposed scheme utilizes two vectors, which are randomly selected, of the top
and bottom 100p% individuals in the current population of size $NP$. The third
vector is randomly selected from the middle [NP-2(100p%)] individuals. A
combination between the mutation rule and the basic mutation strategy
_DE/rand/1/bin_ is applied. It is worth noting that the probability between
the mutation rules is equal ($0.5$). Secondly, a novel self-adaptive scheme
has been introduced that utilizes the gradual change of the crossover rate
values. This scheme, during the evolution process, can benefit from the
previous experience of the individuals in the search space. This feature, on
its turn, can balance effectively the trade-off between the convergence speed
and the population diversity.

###  III-B MLSHADE-SPA

Hadi et al. [12] proposed an LSHADE-SPA memetic framework, which was the
runner up in CEC2018 Large Scale global optimization competition. MLSHADE-SPA
exhibits hybrid characteristics by combining population-based algorithms and
local search. LSHADE-SPA [17], EADE [15], and ANDE [18] are utilized as
population-based algorithms for global exploration, whereas a modified version
of MTS (MMTS) is utilized as a local search algorithm for local exploitation.
In order to further enhance the framework’s performance, the ’divide and
conquer’ concept is applied. This concept is processed without any prior
assumptions of the optimized problems’ structure; the dimensions of the
problems are divided into groups in a random way and each of groups is solved
independently.

###  III-C CBCC-RDG3

A ’divide-and-conquer’ approach is proposed in [13] to address large-scale
optimization problems having components with overlapping behavior. The main
idea behind the ’divide-and-conquer’ approach is to decompose overlapping
problems by modifying the Recursive Differential Grouping (RDG) method. In
order to do so, the linkage to shared by multiple components variables must be
broken. The previously mentioned procedure of decomposition by utilizing RDG
method can be described using the following three steps. The first step is to
distinguish the interaction between the decision variables and the selected
variable $z_{i}$, in order to classify them into a subset $Z_{1}$. The second
step distinguishes and groups the interaction between the decision variables
and any other variable in $Z_{1}$ recursively, in order $Z_{1}$ to becoming
independent from the remaining variables. The third step repeats the two
previous steps until all variables have been grouped. As a result, in the case
of an overlapping problem, an assignment into a single group for all the
decision variables will occur, since all the variables will be linked.

RDG3, which is designed for overlapping problems, modifies the above procedure
by further examining the interaction between the set $Z_{1}$ and the rest of
the available variables (excluding $Z_{1}$) to identify those that interact
with $z_{i}$ indirectly. If the algorithm finds any interaction, the
interacting decision variables are moved to $Z_{1}$. RDG3 repeats these steps
until there is no detection of interaction between $Z_{1}$ and the rest of the
available variables. The decision variables in $Z_{1}$ are considered to be a
consolidated group.

The RDG3 method moves on to the next decision variable $z_{i}$ that has not
been distinguished and grouped. The above process is repeated until all the
decision variables have been classified in groups. RDG3 continues to further
divide the separable variables into small groups with an interval $e_{s}$.
Upon completion of RDG3 method, the identified separable and consolidated
variable groups are returned. The contribution-based CC (CBCC) basic idea is
to allocate available computational resources to components, based on how they
contribute to the overall fitness improvement. This is grouped with RDG3 to
form the CBCC-RDG3 algorithm [13]. The selected solver is the covariance
matrix adaptation- evolutionary strategy (CMA-ES) [19], which utilizes the
round-robin scheme to optimize sub-problems. The computational resources to
each sun-problem, by utilizing the round-robin scheme, are equally
distributed. CBCC-RDG3 was the winner in IEEE CEC 2019 LSGO competition.

###  III-D DGSC-DECC algorithm

The authors in [14] present a differential grouping with spectral clustering
(DGSC) algorithm. The basic concept of Differential Grouping (DG) is to find
the interaction relationship between the available variables according to
their differential values. On the other hand, Spectral Clustering (SC) is
graph partition problem, which is based on spectral graph theory [20], to
provide optimal solutions. Spectral clustering forms an undirected weighted
graph, by considering all the data as edge-connected points in space. The
weight of the edge originates from the similarity value between the two
points. Based on these concepts, the authors in [14] assume that the decision
variables are points in space and consider the interaction relationship
between these variables as the weight of the edge. DGSC combines differential
grouping with spectral clustering, where the similarity matrix of spectral
clustering comes from the differential values of grouping. The final result is
the grouping of the problem’s decision variables. The sub-component optimizer
that is utilized, is described in [21]. This resulted algorithm is called
DGSC-DECC. DGSC-DECC was the runner-up in IEEE CEC 2019 LSGO competition.

##  IV Numerical Results

In this section, the numerical results of the algorithms are presented. We
apply four LSGO algorithms to different WSN configurations, the MLSHADE-SPA
[12], the EADE [15], the CBCC-RDG3 [13], and the DGSC-DECC [14]. For all of
the selected algorithms, we set the population size equal to $100$ and the
maximum number of objective function evaluation to $60,000$. All the selected
algorithms are executed for $50$ independent trials. During the initialization
of each algorithm, the population is randomly selected, based on the lower and
upper boundaries. The number of objective-function evaluations is applied as
the stopping criterion to the optimization problem. The initial position
(solution in the optimization process) of each member of the population in
each dimension is selected within the range [0, 15].

We apply the objective function (cost function) defined in (38). Additionally,
and without losing the generality of the problem, we introduce the assumption
that the coefficients ${{h}_{i}}$, which describe the channel fading, have a
Rayleigh distribution with a unit mean value. Moreover, we also assume that
they are ranked in a descending order
${{h}_{1}}\geq{{h}_{2}}\geq...\geq{{h}_{L}}$.

The corresponding simulations of the first case are performed by applying the
following parameters

  * •

Number of sensor nodes $L=\\{300\\}$,

  * •

Different fusion error threshold values
$\varepsilon=\\{0.001,0.05,0.01,0.1\\}$, and

  * •

Correlation factor values $\rho=\\{0,0.01,0.1,0.5\\}$.

Both values of different fusion error threshold and correlation factor are
applied for each of the sensor nodes. The $SNR$ is set to a fixed value of
$10$dB. Considering all the above, we obtain $4$ different optimization cases
for each $\rho$ value. As a result, the number of dimensions of the total
problem results in 300.

Table I lists the average results of all executions of the algorithms. We
notice that CBCC-RDG3 outperforms the other algorithms in 12 out of the 16
cases. MLSHADE-SPA performs better in four cases having $\varepsilon=0.1$.
However, MLSHADE-SPA obtains a solution in all cases. The other algorithms
fail to produce acceptable solutions and the objective function values are
significantly high.

In the second case, the following parameters are modified

  * •

Number of sensor nodes $L=\\{600,800\\}$,

  * •

Different fusion error threshold values
$\varepsilon=\\{0.001,0.05,0.01,0.1\\}$, and

  * •

Correlation factor $\rho=0$ (Uncorrelated case).

In this case, we set the population size to $250$, whereas the maximum number
of objective function evaluations remains $60,000$. Taking into consideration
the given parameters, the optimization problem becomes more difficult for the
selected algorithms to solve. The comparative results of the algorithms are
listed in Table II. Once again, we notice that CBCC-RDG3 performs better in
six out of the eight cases, however it fails to obtain an acceptable solution
in two cases. MLSHADE-SPA manages to obtain a feasible solution in all cases.
EADE and DGSC-DECC fail to obtain acceptable solutions and achieve higher
objective function values. In this case, the algorithms’ results seem to be
worse than the previous one.

A comparison between the algorithms in terms of two non-parametric statistical
tests is also performed. The rankings according to Friedman test are reported
in Table III. It is noteworthy that the CBCC-RDG3 obtained the best rank of
$1.33$, having the MLSHADE-SPA the second best rank of $1.75$. The Wilcoxon
signed-rank test between the CBCC-RDG3 and the other selected algorithms is
listed in Table IV. The $p$-values below 0.05, which denoted the level of
significance, are indicated in boldface. Again, we can observe that the CBCC-
RDG3 for the optimal power allocation optimization problem is performed
significantly better than all the other algorithms.

The convergence rate plots for $L=\\{300\\}$ and $\rho=\\{0.1,0.5\\}$ are
depicted in Figs. 1 and 2, respectively. We notice that the CBCC-RDG3
converges faster than the other algorithms in all of the given cases except
two. MLSHADE-SPA converges with a similar speed in all cases and achieves a
small value of the objective function. Both algorithms converge at similar
speed at higher objective function values. Moreover, Figs. 3 and 4 portray the
corresponding convergence rate graphs for $L=\\{600,800\\}$ and $\rho=0$. We
can observe that the plots are similar with the previous case. The only
difference is that the CBCC-RDG3 fails to obtain an acceptable solution in two
cases, which is evident by Figs. 3d and 4d.

Concluding our results, it can be clearly seen that MLSHADE-SPA is more stable
than CBCC-RDG3 algorithm with the different fusion error threshold
$\varepsilon=\\{0.001,0.05,0.01,0.1\\}$, as it reaches very promising
solutions in all cases. Besides, having a fusion error threshold
$\varepsilon=\\{0.1\\}$ in all dimensions, MLSHADE-SPA outperforms CBCC-RDG3
algorithm, which proves that MLSHADE-SPA can reach the global solution in
stochastic environment, i.e. it is able to solve very complicated stochastic
real-world problems.

##  V Conclusion

In this paper, we have applied LSGO algorithms to the optimal power allocation
problem in IoT networks. We have compared results with a small number of
objective function evaluations and different numbers of sensors. Overall,
CBCC-RDG3 is a powerful optimizer that obtains good results in most of the
cases. However, MLSHADE-SPA obtains a solution in every case, which could be
very useful in a real-case of wireless communications, where decisions in
real-time are often desirable. EADE and DGSC-DECC failed to obtain acceptable
solutions, which indicates that more objective function evaluations are
probably required. DGSC-DECC performance can be explained since this algorithm
is designed to perform well with overlapping variables, which is not the case
here. In our future work, we will apply LSGO algorithms to more demanding
optimization problems in wireless communications.

## Acknowledgment

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No. 957406 (TERMINET).

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