Evolutionary Large-scale Sparse Multi-objective Optimization for

Collaborative Edge-cloud Computation Ofﬂoading

Guang Peng

, Huaming Wu

, Han Wu

and Katinka Wolter

Department of Mathematics and Computer Science, Free University of Berlin, Takustr. 9, Berlin, Germany

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

Keywords:

Local-edge-cloud, Computation Ofﬂoading, Large-scale Multi-objective Optimization, Restricted Boltzmann

Machine, Contribution Score.

Abstract:

This paper proposes evolutionary large-scale sparse multi-objective optimization (ELSMO) algorithms for col-

laboratively solving edge-cloud computation ofﬂoading problems. To begin with, a collaborative edge-cloud

computation ofﬂoading multi-objective optimization model is established in a mobile environment, where the

ofﬂoading decision is represented as a binary encoding. Considering the large-scale and sparsity property of

the computation ofﬂoading model, the restricted Boltzmann machine (RBM) is applied to reduce the dimen-

sionality and learn the Pareto-optimal subspace. In addition, the contribution score of each decision variable is

assumed to generate new offsprings. Combining the RBM and the contribution score, two evolutionary algo-

rithms using non-dominated sorting and crowding distance methods are designed, respectively. The proposed

algorithms are compared with other state-of-the-art algorithms and ofﬂoading strategies on a number of test

problems with different scales. The experiment results demonstrate the superiority of the proposed algorithms.

1 INTRODUCTION

With the development of mobile networks and the In-

ternet of Things (IoT), more and more complicated

and time-sensitive applications are being developed

for mobile devices (MDs), e.g., face recognition, aug-

mented reality and interactive gaming. Due to the

insufﬁcient computing ability as well as the limited

battery power of MDs, the quality of service (QoS)

of these resource-intensive and delay-sensitive appli-

cations cannot be satisﬁed if they are handled lo-

cally. Fortunately, mobile cloud computing (MCC)

(Sahu and Pandey, 2018) and mobile edge computing

(MEC) (Mach and Becvar, 2017) have emerged for

solving these problems, where the complex comput-

ing tasks can be chosen to be ofﬂoaded to a central

cloud or an edge cloud.

In MCC, taking advantage of more powerful com-

puting capability in a cloud data center, the resource-

intensive tasks can be ofﬂoaded to a central cloud

to alleviate the limitation of computing capability in

MDs. Considering the central cloud may be a lit-

tle far away from MDs, it may cost more time for

transferring the tasks. MEC enables MDs to ofﬂoad

tasks to servers located at the edge of the cellular

network, which reduces the latency of the commu-

nication between MDs and edge servers. Wu (Wu

et al., 2019) proposed a min-cost ofﬂoading partition-

ing (MCOP) algorithm inspired by graph cutting to

minimize the weighted sum of time and energy in

MCC/MEC. Based on his work, Sheikh (Sheikh and

Das, 2018) modeled the effect of parallel execution on

multi-site computation ofﬂoading in MCC and used

an existing genetic algorithm to solve the problem.

Their work focuses on the single-objective optimiza-

tion of ofﬂoading. Some other work used deep learn-

ing methods to solve ofﬂoading problems. Huang

(Huang et al., 2018) formulated the joint ofﬂoading

decision and bandwidth allocation as a mixed inte-

ger programming problem in MEC, and proposed a

distributed deep learning-based ofﬂoading (DDLO)

algorithm for generating ofﬂoading decisions. Yu

(Yu et al., 2020) devised a new deep imitation learn-

ing (DIL)-driven edge-cloud computation ofﬂoading

framework for minimizing the time cost in MEC net-

works. For deep learning methods, one important

concern is that the neural network may need a lot of

time to be trained, and the changing parameters in a

mobile environment can change the structure of the

network. On the other hand, multiple meta-heuristic

optimization algorithms have also gathered attention.

Xu (Xu et al., 2019) applied the evolutionary algo-

rithm NSGA-III (non-dominated sorting genetic al-

gorithm III) to deal with computation ofﬂoading over

100

Peng, G., Wu, H., Wu, H. and Wolter, K.

Evolutionary Large-scale Sparse Multi-objective Optimization for Collaborative Edge-cloud Computation Ofﬂoading.

DOI: 10.5220/0010145501000111

In Proceedings of the 12th International Joint Conference on Computational Intelligence (IJCCI 2020), pages 100-111

ISBN: 978-989-758-475-6

big data for IoT-enabled cloud-edge computing. Guo

(Guo et al., 2018) designed a hierarchical genetic al-

gorithm (GA) and particle swarm optimization (PSO)

algorithm to solve the energy-efﬁcient computation

ofﬂoading management problem.

However, the above methods especially traditional

evolutionary algorithms will encounter difﬁculties to

deal with large-scale computation ofﬂoading prob-

lems when there exists a large number of MDs hav-

ing a huge number of computation tasks. It is difﬁcult

for the traditional evolutionary algorithms to search in

the whole high-dimensional space with limited pop-

ulation size and computational resources. In this

paper, the proposed computation ofﬂoading model

is treated as a multi-objective optimization problem

(MOP) and we focus on multi-objective optimization

evolutionary algorithms (MOEAs). For solving large-

scale MOPs, LMEA (Zhang et al., 2018) divided

the decision variables into convergence-related and

diversity-related variables and optimizes them by dif-

ferent strategies. LCSA (Zille and Mostaghim, 2019)

used a linear combination of population members to

tackle large-scale MOPs. LSMOF (He et al., 2019)

applies two reference points on a solution to search

for better solutions in the large-scale search space.

Although these MOEAs are tailored for large-scale

MOPs, they cannot be applied to MOPs with binary

decision variables and they are shown to be of low ef-

ﬁciency with a large number of function evaluations

(Tian et al., 2020b). Considering some large-scale

MOPs whose Pareto optimal solutions are sparse, the

literature (Tian et al., 2020a) used two unsupervised

neural networks, a restricted Boltzmann machine, and

a denoising autoencoder to learn a sparse distribution

and compact representation of the decision variables,

which is regarded as an approximation of the Pareto-

optimal subspace. In literature (Tian et al., 2020b),

a hybrid representation of solution is adopted to inte-

grate real and binary encodings, which can solve both

large-scale sparse benchmarks and four real applica-

tions well.

Following the above ideas, in this paper we set

up a collaborative edge-cloud computation ofﬂoading

large-scale sparse multi-objective optimization model

with binary encoding. Inspired from literature (Tian

et al., 2020a) and (Tian et al., 2020b), based on the di-

mensionality reduction and decision variable analysis

methods, two evolutionary large-scale sparse multi-

objective optimization algorithms are proposed and

compared to solve the large-scale ofﬂoading prob-

lems. Compared with other MOEAs and ofﬂoading

schemes, the proposed algorithms are competitive on

different large-scale test problems.

The rest of this paper is organized as follows. Sec-

tion 2 brieﬂy introduces the brief background. The

local-edge-cloud ofﬂoading multi-objective optimiza-

tion model is presented in Section 3. The details of the

proposed algorithms are described in Section . The

experimental studies are discussed in Section 4. Fi-

nally, Section 6 summarizes and presents the conclu-

sion and future work.

2 BACKGROUND

In this section some background concepts are pro-

vided.

2.1 Multi-objective Optimization

A multi-objective optimization problem (MOP) can

be deﬁned as follows:

min F(x) = ( f

(x), f

(x),··· , f

(x))

subject to x ∈ Ω ⊆ R

(1)

where Ω is the decision space and x is a solution;

F : Ω → Θ ⊆ R

denotes the m-dimensional objec-

tive vector and Θ is the objective space.

A solution x

is said to Pareto dominate another

solution x

, denoted by x

≺ x

, if







≤ f





, ∀i ∈

{

1,2,· ·· , m

}





< f





, ∃ j ∈

{

1,2,· ·· , m

}

(2)

A solution x

is called a Pareto optimal solution,

if ¬∃x

: x

≺ x

The set of Pareto optimal solutions is deﬁned as

PS =





¬∃x

≺ x



The Pareto optimal solution set in the objective

space is called Pareto front (PF).

2.2 Restricted Boltzmann Machine

Restricted Boltzmann Machine (RBM) is a stochastic

neural network, which consists of an input layer and a

hidden layer, as shown in Fig. 1. The nodes in the two

layers are binary variables obeying binomial distribu-

tion. RBM can be used to reduce the dimensionality

through unsupervised learning. Given an input vector

x, the value of each node h

in the hidden layer is set

to 1 with a probability:

p(h

= 1

x ) = σ

∑

i=1

i j

(3)

where a

is the basis, w

i j

is the weight, D is the di-

mensionality of input layer, σ(x) = 1/(1 + exp (−x))

is the sigmoid function. Through comparing the prob-

ability p (h

= 1

x ) with a uniformly distributed ran-

dom value in [0,1], the binary value of each node h

Evolutionary Large-scale Sparse Multi-objective Optimization for Collaborative Edge-cloud Computation Ofﬂoading

101

can be obtained. In the same way, the reconstructed

value of each node x

in the input layer is set to 1 with

a probability:



= 1



= σ

∑

j=1

i j

(4)

where K is the dimensionality of hidden layer. Hinton

(Hinton, 2002) proposed the contrastive divergence

algorithm to train RBM, which aims to minimize the

reconstruction error between the reconstructed vector

and the original input x by ﬁnding the suitable a, a

and w.

𝑥

bias

𝑥

𝐷−1

𝑥

𝐷

ℎ

𝐾

𝑎

′

𝑤

Figure 1: RBM structure.

3 SYSTEM MODEL AND

PROBLEM FORMULATION

In this section, we consider a collaborative MEC and

MCC network with one edge cloud, one central cloud

and multiple mobile devices (MDs). Thus, the mo-

bile device can execute its computational tasks lo-

cally, or ofﬂoad its tasks to the edge cloud server

through a wireless link and/or to the central cloud

server through wireless and backhaul links.

3.1 System Model

Fig. 2 presents the system model, which consists of

one edge cloud server, one central cloud server and

multiple MDs, denoted by a set N = {1,2, .. ., N}.

The edge cloud server can be deployed into the base

station, which is closer to the MDs. The MDs can

communicate with the edge cloud with a wireless link,

whereas the edge cloud and the central cloud can be

interconnected through a wired link. Each MD has

multiple tasks, denoted by a set M = {1,2, ·· · ,M}.

The size of m-th task of the n-th MD is denoted by

(n,m)

. In each MD, these different tasks can choose

to be processed locally or ofﬂoaded to the edge cloud

server and the central cloud server. The ofﬂoading

decision is represented by two binary variables x

(n,m)

and x

(n,m)

On one hand, x

(n,m)

∈

{

0,1

}

denotes the ofﬂoad-

ing decision for m-th task, which means:

(n,m)



0, if task is executed locally

1, if task is ofﬂoaded

(5)

where x

(n,m)

= 0 denotes n-th MD decides to execute

the m-th task locally, x

(n,m)

= 1 indicates n-th MD de-

cides to ofﬂoad m-th task to the edge cloud server or

central cloud server.

Once the m-th task is decided to be ofﬂoaded to

the cloud server, x

(n,m)

∈

{

0,1

}

represents the speciﬁc

ofﬂoading destination for the m-th task, which means:

(n,m)

(

0, if ofﬂoaded to edge cloud & x

(n,m)

= 1

1, if ofﬂoaded to central cloud & x

(n,m)

= 1

(6)

where x

(n,m)

= 0 denotes n-th MD decides to ofﬂoad

m-th task to the edge cloud server, x

(n,m)

= 1 denotes

m-th task is ofﬂoaded to the central cloud server.

The detailed operations of local computing, edge

cloud computing and central cloud computing are il-

lustrated in Sections 3.2, 3.3 and 3.4, respectively.

The important notations used in this paper are listed

in Table 1.

MD 1

MD 2

MD N

M tasks

M tasks M tasks

Edge cloud

Central cloud

Wireless

Wired

Figure 2: System model of computation ofﬂoading with het-

erogeneous cloud.

3.2 Local Computing Model

We ﬁrst establish the local computing model when the

task is decided to be executed locally. The task exe-

cution time of mobile device can be calculated as:

T l

(n,m)

(7)

where f

denotes the task processing rate of local de-

vice.

ECTA 2020 - 12th International Conference on Evolutionary Computation Theory and Applications

102

Table 1: Important notations used in this paper.

Notation Description

(n,m)

The m-th task workload of the n-th MD

(n,m)

= 0 if m-th task is executed locally, x

(n,m)

= 1 if m-th task is ofﬂoaded to the cloud

(n,m)

= 0 if m-th task is ofﬂoaded to edge cloud, x

(n,m)

= 1 if m-th task is ofﬂoaded to central cloud

(n,m)

The energy consumption of the m-th task of the n-th MD

The local device energy consumption per unit of workload

T l

(n,m)

The execution time of the m-th task of the n-th MD

The task processing rate of the MD

(n,m)

The transmission time of ofﬂoading m-th task to edge cloud via wireless link

(n,e)

The bandwidth between n-th MD and edge cloud

(n,m)

The energy consumption for transmission to the edge cloud

σ The energy consumption per unit of workload for transmission to the edge cloud

T e

(n,m)

The time delay of ofﬂoading the m-th task of the n-th MD to the edge cloud

The task processing rate of the edge cloud

(n,m)

The energy consumption of ofﬂoading the m-th task of the n-th MD to the edge cloud

The energy consumption per unit of workload of edge cloud

(n,m)

The transmission time of ofﬂoading m-th task to central cloud via wireless link and wired link

(e,c)

The bandwidth between edge cloud and central cloud

(n,m)

The energy consumption for transmission to the central cloud

β The energy consumption per unit of workload for transmission to the central cloud

T c

(n,m)

The time delay of ofﬂoading the m-th task of the n-th MD to the central cloud

The task processing rate of the central cloud

(n,m)

The energy consumption of ofﬂoading the m-th task of the n-th MD to the central cloud

The energy consumption per unit of workload of central cloud

The energy consumption for executing m-task at

the n-th device can be calculated as:

(n,m)

= θ

(n,m)

(8)

Where θ

denotes the energy consumption per unit of

workload of local device.

Therefore, the total computation time and energy

consumption of the n-th MD can be expressed as:

T l

(n)

∑

m=1



1 − x

(n,m)



T l

(n,m)

(9)

(n)

∑

m=1



1 − x

(n,m)



(n,m)

(10)

3.3 Edge Cloud Computing Model

For the edge cloud computing model, the MDs can

communicate with the edge cloud via the cellular link.

When the task is decided to be ofﬂoaded to the edge

cloud, the MD needs to transmit the workload of the

task to the edge cloud and then to be processed. In

general, the time and energy consumption are often

neglected when the cloud servers return the comput-

ing results back to MDs, because the data size of feed-

back result is small (Bi and Zhang, 2018).

The transmission time for ofﬂoading the workload

to the edge cloud server can be calculated as:

(n,m)

(n,e)

(11)

where b

(n,e)

denotes the bandwidth between n-th MD

and edge cloud.

The energy consumption for the transmission can

be given by:

(n,m)

= σw(n,m) (12)

where σ denotes the energy consumption per unit of

workload for transmission to the edge cloud server.

After the task is transmitted to the edge cloud, it

will be executed at the edge cloud server. The com-

putation delay of the whole process can be expressed

as:

T e

(n,m)

= T

(n,m)

(13)

where f

denotes the task processing rate of the edge

cloud server.

The energy consumption of whole process can be

expressed as:

(n,m)

= E

(n,m)

+ θ

(n,m)

(14)

where θ

denotes the energy consumption per unit of

workload of edge cloud.

Evolutionary Large-scale Sparse Multi-objective Optimization for Collaborative Edge-cloud Computation Ofﬂoading

103

Therefore, the total computation time and energy

consumption of the n-th MD can be expressed as:

T e

(n)

∑

m=1

(n,m)



1 − x

(n,m)



T e

(n,m)

(15)

(n)

∑

m=1

(n,m)



1 − x

(n,m)



(n,m)

(16)

3.4 Central Cloud Computing Model

For the central cloud computing model, MDs can

communicate with the central cloud via wireless and

wired links. The central cloud servers can provide

more powerful computing capacity for the MDs, but

it might cause more delays for the transmission be-

tween MDs and the central cloud. The transmission

time and energy consumption for ofﬂoading the work-

load to the central cloud server can be calculated as:

(n,m)

(n,e)

(n,m)

(e,c)

(17)

(n,m)

= σw(n,m) + βw (n,m) (18)

where b

(e,c)

denotes the bandwidth between edge

cloud and central cloud. β denotes the energy con-

sumption per unit of workload for transmission to the

central cloud server.

After the task is transmitted to the central cloud, it

will be executed at the central cloud server. The com-

putation delay and energy consumption of the whole

process can be expressed as:

T c

(n,m)

= T

(n,m)

(19)

(n,m)

= E

(n,m)

+ θ

(n,m)

(20)

where f

denotes the task processing rate of central

cloud server, θ

denotes the energy consumption per

unit of workload of central cloud.

Therefore, the total computation time and energy

consumption of the n-th MD can be expressed as:

T c

(n)

∑

m=1

(n,m)

T c

(n,m)

(21)

(n)

∑

m=1

(n,m)

(22)

3.5 Problem Formulation

The total computation time of executing all tasks can

be given by:

T =

∑

n=1

max



T l

(n)

,T e

(n)

,T c

(n)



(23)

The total energy consumption of executing all tasks

can be given by:

E =

∑

n=1



(n)

+ Ee

(n)

+ Ec

(n)



(24)

To summarize, we establish a local-edge-cloud

computation ofﬂoading two-objective optimization

model. Considering a large number of mobile devices

having different applications in the mobile environ-

ment, the ofﬂoading model is the large-scale MOP.

4 THE PROPOSED ALGORITHM

This section presents the details of two proposed al-

gorithms for solving the large-scale computation of-

ﬂoading problems.

4.1 The General Framework

The general frameworks of two proposed algorithms

are presented in Algorithm 1 and 2, respectively. In

both two algorithms, the non-dominated front num-

ber and crowding distance are calculated as the same

way in NSGA-II (Deb et al., 2002), which are also

the selection criteria in the environmental selection.

In ELSMO-1, the non-dominated solutions in the cur-

rent population are used to train a RBM, then the off-

spring solutions are generated from the mating pool.

The two parameters ρ and K are updated iteratively.

In ELSMO-2, the population initialization is differ-

ent from ELSMO-1, which analyzes the contribution

score of each decision variable. And the new off-

spring generation operation is conducted based on

the score. More details about the main operations in

ELSMO-1 and ELSMO-2 are presented in the follow-

ing sections.

4.2 The Proposed ELSMO-1

4.2.1 Offspring Generation

Before generating offspring solutions, all the non-

dominated solutions in the population are used to train

a RBM via the contrastive divergence algorithm. As

shown in Fig. 3, the original binary vectors can be

reduced the dimensionality by RBM, and the reduced

binary vectors can be also recovered. Since the non-

dominated solutions are used to train the RBM, it can

be seen that each solution can be mapped between the

Pareto optimal space and the original search space.

After obtaining the trained RBM, the binary tour-

nament selection mechanism is used to select

N par-

ents as the mating pool based on the non-dominated

ECTA 2020 - 12th International Conference on Evolutionary Computation Theory and Applications

104

Algorithm 1: Framework of ELSMO-1.

Input: The population size

Output: The ﬁnal population P

1 P ← Initialization





;

2 [F

,·· ·] ← NondominatedSorting (P);

3 CrowdDis ← CrowdingDistance (F

,·· ·);

4 ρ ← 0.5; // Ratio of offspring solutions

generated in the different search space

5 K ← N; // Size of the hidden layer

6 while termination criterion not f ul f illed do

7 Train a RBM with K hidden neurons based

on non-dominated solutions in P;

8 P

← Select

N parents via binary

tournament selection in P ;

9 O ←

O f f springGeneration(P,P

,ρ,K,RBM);

10 P ← P ∪ O;

11 Delete duplicated solutions from P;

12 P ← EnvironmentalSelection (P);

13 [ρ,K] ← U pdateParameter (P,ρ) ;

14 end

Algorithm 2: Framework of ELSMO-2.

Input: The population size

Output: The ﬁnal population P

1 [P, Score] ← Initialization





;

2 [F

,·· ·] ← NondominatedSorting (P);

3 CrowdDis ← CrowdingDistance (F

,·· ·);

4 while termination criterion not f ul f illed do

5 P

← Select 2

N parents via binary

tournament selection in P ;

6 O ← O f f springGeneration(P

,Score);

7 P ← P ∪ O;

8 Delete duplicated solutions from P;

9 P ← EnvironmentalSelection (P);

10 end

RBM

Reduce

Recover

Binary

vector

Reduced

binary

vector

Figure 3: Reduce and recover of solutions.

front number and crowding distance. Then two par-

ents are randomly selected from the mating pool and

single-point crossover and bitwise mutation operation

are used to generate two offspring solutions. The ratio

ρ determines the probability that the offspring solu-

tions will be generated in the Pareto-optimal subspace

by the trained RBM or generated in the original search

space. Speciﬁcally, if the probability ρ is larger than

a random value in [0, 1], the binary vectors of parents

are reduced by Eq. (3) and the offspring solutions

are generated in the Pareto-optimal space, then the re-

duced offspring solutions are recovered by Eq. (4).

If the probability ρ is smaller than the random value,

the offspring solutions are generated in the original

search space without RBM.

4.2.2 Update Parameter

In ELSMO-1, there are two parameters needed to

be considered, the ratio of offspring solutions gener-

ated in the Pareto-optimal subspace or original search

space ρ and the size of the hidden layer K. The pa-

rameter ρ is updated iteratively, which is deﬁned as

follows:

t+1

= 0.5 ×



1,t

+ s

2,t



(25)

where ρ

is the value of ρ at the t-th generation and

= 0.5. s

1,t

and s

2,t

denotes the number of suc-

cessful offspring solutions generated in the Pareto-

optimal subspace and in the original search space, re-

spectively. A successful solution means that it sur-

vives to the next population.

The sparsity of non-dominated solutions in the

current population reﬂects the setting value of K. Let

dec be a binary vector (the same dimensionality to

ofﬂoading problem dimension D) denoting whether

each variable should be nonzero, the probability of

setting dec

to 1 is deﬁned according to the non-

dominated solution set NP:

p(dec

= 1

NP ) =

∑

x∈NP

sign(x

)

(26)

where if x

= 0 then

sign(x

)

equals 0 or

sign(x

)

is equal to 1 otherwise. Through comparing the prob-

ability p (dec

= 1

NP ) with a uniformly distributed

random value in [0,1], the value of dec

is obtained.

Then the parameter K is deﬁned as follows:

K =

∑

dec

(27)

Evolutionary Large-scale Sparse Multi-objective Optimization for Collaborative Edge-cloud Computation Ofﬂoading

105

4.3 The Proposed ELSMO-2

4.3.1 The Population Initialization

In ELSMO-2, the population initialization process in-

cludes two steps, i.e., calculating the scores of deci-

sion variables and generating the initial population.

First, the population Q is constituted by a D ∗ D iden-

tity matrix, where D denotes the number of deci-

sion variables. Then the non-dominated front num-

bers of the solutions in population Q are calculated

based on the non-dominated sorting method. The

non-dominated front number of the i-th solution in Q

can be regarded as the score of the i-th decision vari-

able. The higher score means the better quality of the

decision variable, so the value of the decision variable

is set to 1 with higher probability.

Afterwards, let a binary vector mask represent the

ofﬂoading decision. According to the scores of deci-

sion variables, the binary tournament selection mech-

anism is used to select the element in mask with a

better score and set the element to 1. In each solu-

tion, the number of rand () × D elements are selected

to be set to 1 and other elements in mask are set to 0,

where rand () denotes a uniformly distributed random

value in [0, 1]. In this way, the initialized population

with better convergence and diversity is obtained by

selecting decision variables with good quality.

4.3.2 The Offspring Generation

The binary tournament selection mechanism is used

to select 2

N parents as the mating pool based on the

non-dominated front number and crowding distance.

Then two parents p and q are randomly selected from

the mating pool to generate an offspring o each time.

The binary vector mask of o is ﬁrst set to the same to

p, then two decision variables from the nonzero ele-

ments in p.mask ∩ q.

mask are randomly selected with

probability 0.5, the decision variable in the mask of

o with a larger score is set to 0. Otherwise, select-

ing two decision variables from nonzero elements in

p.mask ∩ q.mask, the decision variable in the mask of

o with smaller score is set to 1.

Afterwards, one mutation operation is conducted

on the mask of o to retain diversity. Compare one ran-

dom probability distributed in [0,1] with 0.5, if the

probability is less than 0.5, randomly select two deci-

sion variables from the nonzero elements in o.mask,

and set the element with large score in o.mask to

0. Otherwise, randomly select two decision variables

from the nonzero elements in o.mask, and set the ele-

ment with a small score in o.mask to 1. The main idea

of the mutation operation is making decision variables

with better quality approach to 1, while decision vari-

ables with worse quality approach 0.

4.4 Computational Complexity

For the proposed algorithms, the major costs are the

iteration process in Algorithm 1 and 2. In ELSMO-

1, Step 7 needs O



NEDK



operations to train the

RBM, where

N is the population size, E is the num-

ber of epochs for training, D is the number of de-

cision variables, K is the hidden layer size. Step 8

needs O





operations for the binary tournament se-

lection. Step 9 performs O



NDK



to generate off-

spring solutions. Step 12 performs O





opera-

tions for the environmental selection, where

M is the

number of objectives. Step 13 needs O





oper-

ations to update the parameters. To summarize, the

overall computational complexity at one generation

of ELSMO-1 is max



NEDK





o

. In

ELSMO-2, Step 5 performs O





operations for

selecting mating pool. Step 6 needs O





opera-

tions to generate offspring solutions. Step 9 performs





operations for the environmental selection.

The overall computational complexity at one genera-

tion of ELSMO-2 is max







o

NSGA-II (Deb et al., 2002), SPEA2 (Zitzler et al.,

2001), SMS-EMOA (Hochstrate et al., 2007), and

EAG-MOEA/D (Cai et al., 2015) are four compared

algorithms selected in this paper. The computational

complexity of NSGA-II, SMS-EMOA and EAG-

MOEA/D is the same, i.e. O





. The worst com-

putational complexity of SPEA2 is O





N +





where N is the archive size.

5 EXPERIMENTAL STUDIES

In this section, we evaluate the performance of the

proposed algorithms and demonstrate the compared

results of different MOEAs and ofﬂoading strategies.

5.1 Experimental Settings

In the experiment, we set up the local-edge-cloud of-

ﬂoading environment. The number of mobile devices

is selected between 100 and 1000. The number of

ECTA 2020 - 12th International Conference on Evolutionary Computation Theory and Applications

106

independent tasks of each MD is M = 5. So the di-

mension of the ofﬂoading problem D is between 1000

and 10000. We set the energy consumption per unit

of workloads in local device, edge and central cloud

server θ

= 3J/MB, θ

= 1.5J/MB, and θ

= 1J/MB,

respectively. The processing rates of local device,

edge and central cloud server are f

= 2MB/s, f

8MB/s, and f

= 12MB/s, respectively. The energy

consumption per unit of workloads for transmission

from local device to edge cloud server is a random

value within [0.4, 0.6] J/MB, and the same measure

from edge to central cloud server is random value

within [0.3,0.5] J/MB. In addition, the bandwidth be-

tween the local device and edge server is chosen from

[80,100] Mbps, whereas the bandwidth between edge

cloud and central cloud is ﬁxed b

(e,c)

= 150Mbps. We

assume that the workloads of tasks are randomly dis-

tributed between 10MB and 30MB. The related pa-

rameters and corresponding values are summarized in

Table 2.

To verify the performance of the proposed algo-

rithms, we compare the proposed algorithms with

other MOEAs and ofﬂoading schemes to solve

ten different large-scale ofﬂoading problems, which

means that the number of devices N = [100, 200, 300,

400, 500, 600, 700, 800, 900, 1000] and the dimen-

sion D = [1000, 2000, 3000, 4000, 5000, 6000, 7000,

8000, 9000, 10000]. The compared algorithms are

NSGA-II (Deb et al., 2002), SPEA2 (Zitzler et al.,

2001), SMS-EMOA (Hochstrate et al., 2007), and

EAG-MOEA/D (Cai et al., 2015). NSGA-II, SPEA2,

SMS-EMOA are three classical MOEAs which are

effective for MOPs, while EAG-MOEA/D is tailored

for combinatorial MOPs. For a fair comparison, the

population size of all algorithms is set to 50. The

number of function evaluations is set with values in

the interval from 6.0 × 10

to 15 × 10

for ten test

problems, and the interval of the number of func-

tion evaluations is 10

. In NSGA-II, SPEA2, SMS-

EMOA, EAG-MOEA/D and ELSMO-1, the single-

point crossover and bitwise mutation are applied to

generate new offspring, where the probabilities of

crossover and mutation are set to 1.0 and 1/D, respec-

tively. The hypervolume (HV) (Zitzler et al., 2003) is

adopted as the metric to evaluate the performance of

the compared algorithms. The HV can reﬂect the per-

formance regarding both convergence and diversity.

The bigger the HV value, the better the performance

of the algorithm. For each test instance, each algo-

rithm is executed 30 times independently, and the av-

erage and standard deviation of the metric values are

recorded. The Wilcoxon rank sum test at a 5% signif-

icance level is used to compare the experimental re-

sults, where the symbol ’+’, ’−’ and ’≈’ denotes that

the result of another compared algorithm is signiﬁ-

cantly better, signiﬁcantly worse and similar to that

obtained by the proposed algorithm.

The other four compared ofﬂoading schemes are

Local Ofﬂoading Scheme (LOS), Edge Ofﬂoading

Scheme (EOS), Cloud Ofﬂoading Scheme (COS),

Random Ofﬂoading Scheme (ROS). LOS, EOS, and

COS represent that all applications are executed on

the local device, ofﬂoaded to edge and central cloud.

ROS denotes that the ofﬂoading decisions are gener-

ated randomly. To demonstrate the effectiveness of

different ofﬂoading schemes, the ofﬂoading gain of a

weighted sum of time and energy based on LOS is

deﬁned as:

O f f loadingGain =

w ×

LOS

−T

o f f loading

LOS

+ (1 − w)

LOS

−E

o f f loading

LOS

× 100%

(28)

where T

LOS

and E

LOS

denote the time and energy

consumption of LOS, respectively. T

o f f loading

and

o f f loading

denotes the time and energy consumption

of other ofﬂoading schemes. w is the weight parame-

ter, which can be set by the decision-maker.

5.2 Comparison with Other MOEAs

Table 3 presents the HV metric values obtained

by NSGA-II, SPEA2, SMS-EMOA, EAG-MOEA/D,

ELSMO-1 and ELSMO-2 on ten test problems. The

proposed ELSMO-2 has achieved the best perfor-

mance on 9 of 10 test instances, while only EAG-

MOEA/D gets 1 of 10 best results for the rest of com-

pared algorithms. When the dimension is not so large

(i.e., 1000), the EAG-MOEA/D and ELSMO-2 can

obtain similar metric values. It can be observed that

ELSMO-2 has a clear advantage over other compared

algorithms with the increment of dimension.

Fig. 4, Fig. 5 and Fig. 6 show the ﬁnal non-

dominated solution set with the medium HV value

obtained by NSGA-II, SPEA2, SMS-EMOA, EAG-

MOEA/D, ELSMO-1 and ELSMO-2 on ofﬂoading

problems with 1000, 5000, and 10000 binary vari-

ables. NSGA-II, SPEA2, SMS-EMOA can only get

a small part of the solutions in the Pareto front, which

may be worse when dealing with large-dimension

problems. EAG-MOEA/D is designed for combina-

torial problems, which can obtain good performance

compared with other classical algorithms NSGA-II,

SPEA2 and SMS-EMOA, whereas its diversity still

encounters difﬁculties for solving large-dimensional

ofﬂoading problems. ELSMO-1 seems to have the

best performance of diversity compared with the other

ﬁve algorithms, but the RBM may need more itera-

tions to be trained for improving the convergence. It

Evolutionary Large-scale Sparse Multi-objective Optimization for Collaborative Edge-cloud Computation Ofﬂoading

107

Table 2: Parameter values.

Parameter Value

The number of mobile devices N = [100,1000]

The number of independent tasks of each MD M = 5

The local energy consumption per unit θ

= 3J/MB

The edge cloud energy consumption per unit θ

= 1.5J/MB

The central cloud energy consumption per unit θ

= 1J/MB

The processing rate of the local device f

= 2MB/s

The processing rate of the edge cloud server f

= 8MB/s

The processing rate of the central cloud server f

= 12MB/s

The energy consumption per unit for transmission from MD to edge cloud σ = [0.4,0.6]J/MB

The energy consumption per unit for transmission from edge to central cloud β = [0.3,0.5] J/MB

The bandwidth between n-th MD and edge cloud b

(n,e)

∈ [80,100]Mbps

The bandwidth between edge and central cloud b

(e,c)

= 150Mbps

The workloads of all tasks w

(n,m)

∈ [10,30]MB

Figure 4: The non-dominated solution set with the medium HV value obtained by NSGA-II, SPEA2, SMS-EMOA, EAG-

MOEA/D, ELSMO-1 and ELSMO-2 on the 1000-dimensional ofﬂoading problem.

is clear from the ﬁgures that ELSMO-2 can always get

better performance between convergence and diver-

sity no matter what the dimensional ofﬂoading prob-

lem.

5.3 Comparison with Other Ofﬂoading

Schemes

It has been observed that ELSMO-2 can get the best

non-dominated solution set in the Pareto front. The

non-dominated solution set can give the decision-

maker more choices. To further validate the perfor-

mance of ELSMO-2, we compare the ofﬂoading gain

between ELSMO-2 and EOS, COS and ROS based

on the LOS. According to the different quality of ser-

vice, the decision-maker may set different weights of

w for the tradeoff between time and energy. If the

decision-maker is sensitive to the time consumption,

it may set a larger weight for the time consumption, or

if the decision-maker focus is on energy performance,

it may set a larger weight for the energy consumption.

Fig. 7, Fig. 8 and Fig. 9 presents the ofﬂoading

gain of different ofﬂoading schemes under the differ-

ent weights on 1000-, 5000-, 10000-dimensional of-

ﬂoading problems. It can be seen that ofﬂoading is al-

ways beneﬁcial compared with the only local ofﬂoad-

ing scheme. And the proposed ELSMO-2 can always

get the best ofﬂoading gain compared with other of-

ﬂoading schemes under different weights on different

large-scale ofﬂoading problems. On the other hand,

ECTA 2020 - 12th International Conference on Evolutionary Computation Theory and Applications

108

Figure 5: The non-dominated solution set with the medium HV value obtained by NSGA-II, SPEA2, SMS-EMOA, EAG-

MOEA/D, ELSMO-1 and ELSMO-2 on the 5000-dimensional ofﬂoading problem.

Figure 6: The non-dominated solution set with the medium HV value obtained by NSGA-II, SPEA2, SMS-EMOA, EAG-

MOEA/D, ELSMO-1 and ELSMO-2 on the 10000-dimensional ofﬂoading problem.

ELSMO-2 takes an obvious advantage over other of-

ﬂoading schemes when w is becoming larger, which

means that ELSMO-2 can reduce the time delay more

efﬁciently. What’s more, compared with ROS, EOS

and COS can obtain a better ofﬂoading gain, which

also demonstrates that edge and cloud ofﬂoading can

improve performance for both time and energy con-

sumption.

6 CONCLUSION AND FUTURE

WORK

In this paper, two evolutionary large-scale sparse

multi-objective optimization (ELSMO) algorithms

are proposed and compared for solving heterogeneous

edge-cloud computation ofﬂoading problems. Con-

sidering the large-scale and sparsity properties of the

multi-objective ofﬂoading model, the RBM is used to

reduce the dimensionality and learn from the Pareto

Evolutionary Large-scale Sparse Multi-objective Optimization for Collaborative Edge-cloud Computation Ofﬂoading

109

Table 3: The average HV values obtained by NSGA-II, SPEA2, SMS-EMOA, EAG-MOEA/D, ELSMO-1 and ELSMO-2 on

ofﬂoading problems. Highlighted values corresponds to the best results according to the statistical tests.

Problem D NSGA-II SPEA2 SMS-EMOA EAG-MOEA/D ELSMO-1 ELSMO-2

Ofﬂoading1

1000 2.7834e-1 (1.60e-3) − 2.7353e-1 (1.75e-3) − 2.6961e-1 (1.64e-3) − 2.9834e-1 (3.45e-4) + 2.9541e-1 (8.36e-4) − 2.9800e-1 (8.96e-5)

Ofﬂoading2

2000 2.6303e-1 (1.80e-3) − 2.5928e-1 (1.20e-3) − 2.5669e-1 (9.36e-4) − 2.9564e-1 (9.80e-4) − 2.6861e-1 (2.92e-2) − 2.9761e-1 (1.73e-4)

Ofﬂoading3

3000 2.5533e-1 (1.70e-3) − 2.5192e-1 (9.69e-4) − 2.5123e-1 (1.44e-3) − 2.8934e-1 (3.76e-3) − 2.8580e-1 (2.07e-3) − 2.9442e-1 (3.48e-4)

Ofﬂoading4

4000 2.5263e-1 (1.10e-3) − 2.4856e-1 (7.96e-4) − 2.4805e-1 (7.14e-4) − 2.8371e-1 (3.79e-3) − 2.7427e-1 (1.89e-2) − 2.9761e-1 (1.73e-4)

Ofﬂoading5

5000 2.5009e-1 (1.09e-3) − 2.4757e-1 (6.41e-4) − 2.4656e-1 (8.32e-4) − 2.8441e-1 (1.35e-3) − 2.7836e-1 (3.87e-3) − 2.9161e-1 (3.14e-4)

Ofﬂoading6

6000 2.4780e-1 (6.85e-4) − 2.4587e-1 (6.99e-4) − 2.4483e-1 (4.67e-4) − 2.8219e-1 (2.73e-3) − 2.7564e-1 (2.51e-3) − 2.9018e-1 (2.00e-4)

Ofﬂoading7

7000 2.4759e-1 (1.04e-3) − 2.4493e-1 (1.59e-4) − 2.4391e-1 (2.20e-4) − 2.7632e-1 (5.08e-3) − 2.6431e-1 (1.77e-2) − 2.8837e-1 (5.94e-4)

Ofﬂoading8

8000 2.4542e-1 (6.92e-4) − 2.4373e-1 (7.42e-4) − 2.4254-1 (6.27e-4) − 2.7476-1 (5.20e-3) − 2.7134e-1 (1.67e-3) − 2.8626e-1 (8.37e-4)

Ofﬂoading9

9000 2.4439e-1 (4.68e-4) − 2.4325e-1 (3.76e-4) − 2.4218-1 (7.11e-4) − 2.7607e-1 (3.52e-3) − 2.6904e-1 (2.81e-3) − 2.8521e-1 (3.74e-4)

Ofﬂoading10

10000 2.4434e-1 (3.23e-4) − 2.4267e-1 (5.62e-4) − 2.4212-1 (3.56e-4) − 2.7024e-1 (7.04e-4) − 2.7013e-1 (2.49e-3) − 2.8359e-1 (1.17e-3)

+/ − / ≈ 0/10/0 0/10/0 0/10/0 1/9/0 0/10/0

(a) D = 1000 (b) D = 5000 (c) D = 10000

Figure 7: Ofﬂoading gain of different ofﬂoading schemes for w = 0.2.

(a) D = 1000 (b) D = 5000 (c) D = 10000

Figure 8: Ofﬂoading gain of different ofﬂoading schemes for w = 0.5.

(a) D = 1000 (b) D = 5000 (c) D = 10000

Figure 9: Ofﬂoading gain of different ofﬂoading schemes for w = 0.8.

optimal subspace. While the contribution score is

applied to select better decision variables to gener-

ate offspring. The proposed algorithms are compared

with other MOEAs and ofﬂoading schemes to solve

the test problems under different scales. The experi-

mental results demonstrate the effectiveness and efﬁ-

ciency of the proposed algorithms.

In the future, some other efﬁcient methods that

can reduce the dimensionality will be considered,

such as Principal Component Analysis (PCA) and

ECTA 2020 - 12th International Conference on Evolutionary Computation Theory and Applications

110

Autoencoder (AE). And also, the relationship be-

tween different decision variables will be investi-

gated.

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