Surrogate models for multi-objective evolutionary algorithms, a survey and current research trends

Công nghệ thông tin 
N. D. Dinh, , N. X. Hoai, “Surrogate models for multi-objective  research trends.” 120 
SURROGATE MODELS FOR MULTI-OBJECTIVE 
EVOLUTIONARY ALGORITHMS, A SURVEY 
AND CURRENT RESEARCH TRENDS 
Nguyen Duc Dinh1, Nguyen Long2*, Nguyen Hong Nghi3, Nguyen Xuan Hoai4 
Abstract: Multi-objective problems (MOPs), a class of optimization problems in 
the real-word, which has multiple conflicting objectives. Multi-objective 
evolutionary algorithms (MOEAs) are known as great potential algorithms to solve 
difficult MOPs. With MOEAs, based on principle of population, we have a set of 
optimal solutions (feasible solution set) after the search. We often use concept of 
dominance relationship in population, it is not difficult to find out set of Pareto 
optimal solutions during generations. However, with expensive optimization 
problems in the real world, it has to use a lot of fitness function evaluations during 
the search. To avoid the expensive physical experiments, we can use computer 
simulations methods to solve the difficult MOPs. In fact, this way often costs 
expensive in computation and times for the simulation. In these cases, researchers 
discussed on the usage of surrogate models for evolutionary algorithms, especially 
for MOEAs to minimize the number of fitness callings. This paper we will take a 
short overview about surrogate models for MOEAs, state of the art and current 
research trends. 
Keywords: Surrogate models; Approximation models; Meta-models; MOEAs. 
1. COMMON CONCEPTS 
1.1. Expensive multi-objective problems 
A multi-objective optimization problem involves at least two conflicting 
objectives and it has a set of Pareto optimal solutions. MOEAs use a population of 
solutions to approximate the Pareto optimal set in a single run. MOEAs have 
attracted a lot of research attention during the past decade. They are still one of the 
hottest research areas in the field of Computational Intelligence, especially to solve 
difficult MOPs in the real world. 
 A multi-objective problem is formed as follows [16]: 
minimize {f1(x), f2(x), , fk(x)} (1) 
 subject to x S, 
where k ( 2) is the number of objectives, fi: R
n R are objective functions. The 
vector of objective functions are denoted by f(x) = (f1(x), f2 (x),..., fk(x))
T. The decision 
(variable) vector x = (x1, x2,..., xn)
T belongs to the feasible region (set) S, which is a 
subset of decision variable space Rn. The term “minimize” means all objective 
functions are minimized simultaneously. 
If there is no conflict between objective functions, then a solution can be found 
where every objective function attains its optimum. In this case, no special 
methods are needed. To avoid such trivial cases we assume that there does not 
exist a single solution that is optimal with respect to every objective function. This 
means that the objective functions are at least partly conflicting. 
In multi-objective optimization, a problem is defined as a expensive problem 
when it has fitness functions which require time-consuming experiments or 
computer simulations. 
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1.2. Surrogate models 
Surrogate models are used to approximate in simulation way to reduce the 
computational cost for expensive problems. The models are described as below: 
If we call ( )f x
 
 is an origin fitness function of a MOP, then, we have '( )f x
 
is a 
meta function, which is indicated as below: 
'( ) ( ) ( )f x f x e x 
   
 (2) 
function ( )e x
 
is the approximated error. In this case, the fitness function ( )f x
 
is not to be known, the values (input or output) are cared. Based on the responses 
of the simulator from a chosen dataset, a surrogate is constructed, then the model 
generates easy representations that describe the relations between preference 
information of input and output variables. There are some approaches for the 
surrogate models, which are divided into some kinds as below: 
1.2.1. The radial basis function (RBF) 
In the proposal [8], the authors suggest an approach to develop equations of 
topography and other irregular surfaces. To describe the method, the term “multi 
quadric analysis” is used. The radial basis function (RBF) is a real value function 
which is depended on the distance from a center point of the neuron to the input 
point. A specified point in the neuron may used as the center point. The radial 
function is formed as below: 
 ( )x x  
 (3) 
The RBF often includes of three different layers, such as: 
• Input layer: works on an identify function. 
• Hidden layer: works on non-linear RBF activation function. 
• Output layer: works on 1: ( )
Nn
i ii
R R x w x c  
 
  
In this approach, some parameters are used: the central vector i
c
 
, the weights wi 
and the RBF widths i . The optimization process is the task of tuning these 
parameters. Based on the model, there are some proposals recently [11, 26, 5, 17], 
for details: 
In [11], the authors proposed an algorithm which performs actual analysis for 
the initial population and periodically every few generations. An external archive 
of the unique solutions evaluated using the actual analysis is maintained to train 
the surrogate models. The data points in the archive are split into multiple 
partitions using k-Means clustering. A RBF network surrogate model is built for 
each partition using a fraction of the points in that partition. The rest of the points 
in the partition are used as a validation data to decide the prediction accuracy of the 
surrogate model. Prediction of a new candidate solution is done by the surrogate 
model with the least prediction error in the neighborhood of that point. 
The authors in [26] introduced a modified version of MOEA/D which is assisted 
by cooperative RBF networks, with the aim of improving the prediction of the 
function value. The RBF networks employed here, use different kernels in order to 
have different shapes of the fitness landscape. With that, each RBF network 
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N. D. Dinh, , N. X. Hoai, “Surrogate models for multi-objective  research trends.” 122 
provides information which is used to improve the value of the objective function. 
In [14] the authors presented a meta-model assisted memetic algorithm, using 
online trained local meta-models supporting both the inexact pre-evaluation of 
evolving populations and the local search. The use of RBF networks to screen out 
non-promising individuals in each generation, so as to avoid evaluating them with 
the exact and costly evaluation tool, is currently a well established technique; it 
considerably reduces the CPU cost of evolutionary computations and makes them 
competitive for engineering applications with computationally expensive 
evaluation tools. 
In [17], the authors proposed a new MOEA called MODE-LD+SS, which 
combines differential evolution with local dominance and scalar selection 
mechanisms. Local dominance aims to improve the convergence rate and the scalar 
selection mechanism intends to improve the distribution of solutions along the 
Pareto front. 
1.2.2. The polynomial response surface (PRS) 
The authors in [21] suggested using concept of statics to regress and analyst the 
variances to find out the minimum responsive variance. This methodology called 
The response surface methodology (RSM). Based on the RSM and polynomials, 
and popular approach is proposed, which called polynomial response surfaces 
(PRS). In PRS, a function is a linear aggregate (or combination) of powers and 
products of the training set. The model is formed as below: 
( )ˆ
pp Ty x 
 (4) 
Here, dataset sized n is presented as 1 2, ,..., ,nx x x
   
 β is the vector of coefficients 
to be estimated, 
p
x
 is the vector corresponding to the form of a pair of x1
(p) and 
x2
(p). 
In the proposal, there are two common methods are used to estimate unspecified 
coefficients: gradient and least squares methods. In this case, the number of 
coefficients is the number of samples which we need. 
In [3], the authors used stepwise regression to build up PRS as below: 
• Determine an initial model. 
• In a loop, the model is altered with one of two actions (add or remove a 
predictor variable in accordance with a special criterion. 
• Stop the search when the loop size is reached. 
Based on the approach, there are some proposals in [6, 13, 4, 20,18], for details: 
Based on NSGA-II, the authors on [6] introduced a proposal which applied to 
study trade-offs among objectives of a rocket injector design problem where 
performance and life objectives compete. The Pareto Optimal Front (POF) is 
approximated using a quintic polynomial. The compromise region quantifies trade-
offs among objectives. 
In [13], to solve the problem on crash safety design of vehicles, the authors 
suggest to use stepwise regression model for NSGA-II. The authors present a 
multi-objective optimization procedure for the vehicle design, where the weight, 
acceleration characteristics and toe-board intrusion are considered as the design 
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objectives. The response surface method with linear and quadratic basis functions 
is employed to formulate these objectives, in which optimal Latin hypercube 
sampling and stepwise regression techniques are implemented. 
The authors in [4] presented and analyzed in detail an efficient search method 
based on evolutionary algorithms (EA) assisted by local Gaussian random field 
meta-models (GRFM). It is created for the use in optimization problems with one 
(or many) computationally expensive evaluation function(s). The role of GRFM is 
to predict objective function values for new candidate solutions by exploiting 
information recorded during previous evaluations. Moreover, GRFM are able to 
provide estimates of the confidence of their predictions. Predictions and their 
confidence intervals predicted by GRFM are used by the meta-model assisted EA. 
It selects the promising members in each generation and carries out exact, costly 
evaluations only for them. The extensive use of the uncertainty information of 
predictions for screening the candidate solutions makes it possible to significantly 
reduce the computational cost of single and multi-objective EA. 
In [18], the authors used a special memetic operator, which performs local 
search on some of the newly generated individuals. This operator used the meta-
model constructed based on previously evaluated points in the decision space. The 
meta-model is trained to predict the distance to the currently known Pareto front. 
In the proposal, the known points do not have the same weight, as those that are 
closer to the locally optimized one are considered more important. The author’s 
idea is that points closer to the Pareto front are more interesting during the run of 
the algorithm, and the memetic operator moves the individuals towards the front. 
The meta-model provides a general direction in which the search should proceed. 
To obtain a training set for the meta-models we also added an external archive of 
individuals with known objective values. 
A meta-model based approach is introduced in [20] to the reduction in the 
needed number of function evaluations is presented. Local aggregate meta-models 
are used in a memetic operator. The algorithm is first discussed from a theoretical 
point of view and then it is shown that the meta-models greatly reduce the number 
of function evaluations. 
1.2.3. The support vector machine (SVM) 
In [24], based on the theory of statistical learning, the authors introduced to use 
support vector machines (SVMs) which is a set of related supervised learning 
methods. Here, the data is analyzed to recognize patterns. In this approach, to 
classify, regress, analyst one or set of hyper-planes in corresponding multi-
dimensional space is suggested to use. The set of inputs are map to a larger space, 
then, it calculates the cross product in terms of the variables in original space 
which makes the computational load reasonable. The concept of “kernel function” 
(as K(x, y)) is defined as the cross products in larger space. The kernel function 
can be chosen as a solver for the regression problem. One of other function is 
introduced is “loss function” with a measure of distance. The problem of 
approximating the set of data is formed as: 
 , wf x g x b (5) 
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Then, the minimum of the function: 
 2
1
1
( , )
2
n
i i
i
w w C    
  (6) 
is known as the optimal regression function. Here, C is specified value,  and 
 are slack variables which present the lower and upper constraints on the results. 
Based on the SVM, there are some proposals in [14, 19, 20], for details: 
In [14] the authors proposed a meta-model based approach to the reduction in 
the needed number of function evaluations is presented. Local aggregate meta-
models are used in a memetic operator. The algorithm is first discussed from a 
theoretical point of view and then it is shown that the meta-models greatly reduce 
the number of function evaluations. 
In [19], the authors proposed a surrogate-based evolutionary strategy for multi-
objective optimization. The evolutionary strategy uses distance based aggregate 
surrogate models in two ways: as a part of memetic search and as way to pre-select 
individuals in order to avoid evaluation of bad individuals. The model predicts the 
distance of individuals to the currently known Pareto set. 
In [20], the authors presented a memetic evolutionary algorithm for multi-
objective optimization with local meta-models. The proposal shows that the local 
models give better results than a single global model, usually reducing the number 
of needed function evaluations by 10%, with occasional reductions as high as 48%. 
Although this difference may seem rather small, it may better reduce the associated 
costs in practical tasks. 
1.2.4. The kriging (KRG) 
In [15], the authors proposed a method named “Kriging”, which is a response 
surface method bases on spatial prediction techniques. This method minimizes the 
mean squared error to build the spatial and temporal correlation among the values 
of an attribute. In [22], the authors developed a parametric regression model which 
design and analysis of computer experiments, called DACE. The model is an 
extension of Kriging approach for at least three dimensions problems. The model 
is a combination of a known function f(x) and a Gaussian random process f’(x) that 
is assume to have mean zero and covariance, like as: 
( ) ( ) ( ) ( ) ( ) ( )2( '( ), '( )) ( '( ), '( )) ( , , )
i j  ... networks have advantages of easy design, good generalization, 
strong tolerance to input noise, and online learning ability. The properties of RBF 
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networks make it very suitable for design problems in the real-world. The RBF is 
training faster than multi-layer perceptron (MLP). In RBF, the hidden layer is 
easier to interpret than the hidden layer in an MLP. 
- PRS: The integration of multiple disciplinary codes with an optimization 
code becomes more manageable. The statistical methods can be used to detect 
and repair bad data and to estimate the average error in the data [9]. The model is 
suited to simple design landscapes with low dimensionality where data is cheap 
to obtain [23]. 
- SVM: With the concept of kernel, SVMs gain flexibility in the choice of the 
form of the threshold separating solvent from insolvent companies, which needs 
not be linear and even needs not have the same functional form for all data, since 
its function is non-parametric and operates locally. In the model, by choosing an 
appropriate generalization grade, SVMs can be robust, even when the training 
sample has some bias. By choosing different r values for different input values, it 
is possible to rescale outliers. SVMs deliver a unique solution, since the optimality 
problem is convex. With appropriate kernel, such as the Gaussian kernel, one can 
put more stress on the similarity between companies, because the more similar the 
financial structure of two companies is, the higher is the value of the kernel. 
- Kriging: Kriging treats clusters as like as the point's help to carry estimate the 
effects of data clustering. With the model the known value of estimation errors 
(kriging variance) along with the z variable and it gives estimation of errors for 
stochastic simulation of possible realizations of Z(u) [12]. 
2.2. Disadvantages 
The models for surrogate assisted MOEAs show a lot of advantages in 
optimization area, but they also give some disadvantages: 
- RBF: Although the RBF is quick to train, when training is finished and it is 
being used it is slower than a MLP, so where speed is a factor a MLP may be more 
appropriate. 
- PRS: Care is needed with the selection of the polynomial order. For problems 
with a high number of dimensions, it is difficult to gain sufficient data. Care is 
needed with order selections. 
- SVM: the biggest limitation of the SVM model which lies in choice of the 
kernel, speed and size, both in training and testing, high algorithmic complexity 
and extensive memory requirements of the required quadratic programming in 
large-scale tasks. 
- Kriging: This model is expensive to train for higher dimensional problems. 
3. CURRENT RESEARCH TRENDS 
Through a short review and analysis about the common models for surrogate 
multi-objective evolutionary algorithm, recently proposals, research issues and 
discussions, there are some promised trends for academic researchers in this area: 
1. Surrogates in interactive evolutionary multi-objective (EMO) algorithms: 
in an interactive EMO algorithm, humans calculate the fitness value for all 
individuals. This is necessary in case of no fitness functions are available. It is 
difficult for the algorithm when humans feel tired, have no choices, no actions. To 
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solve the problem, the usage of surrogate models can be used to replace the 
evaluations by the human. This case, machine learning model is used to predict the 
fitness value the human may assign to a design based on history data. 
2. Surrogated-assisted evolution for solving dynamic multi-objective 
optimization algorithms: Recently, researchers discussed a lot about evolutionary 
algorithms for dynamic MOPs. The research idea is to design an evolutionary 
search strategy to drive solutions belong POF. In this, the maintaining of diversity 
of population is an important part of the evolutionary algorithm. Then, memory 
mechanisms which include sub-populations, archives, external population need to 
construct as research directions. To implement the strategy to maintain the 
diversity of the population anticipation and prediction of the change in the fitness 
function can be helpful in solving dynamic problems more efficiently. In such 
strategies, a surrogate can be helpful in learning the changing fitness function. 
3. Surrogates for robust multi-objective optimization: In the real-world 
applications, we need to concern the performance of MOEAs to obtain optimal 
solution and also the sensitivity of the performance to small. If an optimal solution 
is insensitive to such changes, the solution is known as robust optimization. This 
case, an implicit averaging or explicit averaging can be used in MOEAs to obtain 
robust optimal solutions. Wherein an assumption on the probability distribution of 
the noise is often made. By contrast, one can predefine the allowed performance 
decrease and then search for an optimum that has the maximum tolerance of 
changes in the design variables or in the environment, which is termed inverse 
robust optimization. In both cases, explicit averaging based or inverse robust 
optimization additional fitness evaluations are needed. Then, a surrogate is 
promised direction to improve the efficiency and extra fitness evaluations. 
4. Surrogate-assisted combinatorial multi-objective optimization: In the real-
world applications, there are also many computationally intensive combinatorial 
optimization problems. In these cases, we can use discrete modeling techniques 
such as binary neural network, which is used to assist a mixed integer evolution 
strategy for medical images analysis. Then, the Kriging model is suitable to be 
used for telecom network optimization problems. 
5. Surrogates for constrained optimization: In the real-world, MOPs often 
have one or more constraints. To determine if a given solution is feasible, we need 
to evaluate the constraint functions. In case of the evaluations of constraint 
functions are time consuming, it is desirable to replace the constraint functions 
with others by using approximate models. Then, surrogates are applied to 
manipulate the shape and size of the feasible region to ease the solution of highly 
constrained MOPs. The usage of surrogate models to deliberately enlarge the 
feasible region by building up a very simple surrogate for each constraint function. 
As the optimal proceeds, the complexity of the surrogates increases gradually so 
that the approximated feasible region can converge to the real feasible region. 
4. CONCLUSION 
The usage of surrogate models for multi-objective evolutionary algorithms are 
motivated from real-world applications. As multi-objective optimizations are 
Công nghệ thông tin 
N. D. Dinh, , N. X. Hoai, “Surrogate models for multi-objective  research trends.” 128 
increasingly applied to solving complex problems, expensive problems research 
trends in using surrogate models for MOEAs have considerably increased in recent 
years. This paper provides a brief overview of recent advances in this research area 
and raised some research trends that remain to be resolved in the future. Based on 
research trends, researchers need to apply new techniques to incorporate with the 
surrogate models for MOEAs such as Artificial Intelligence (AI), Deep Learning, 
Cloud Computing,... with which more computing resources will be made available 
to common users via computer networks. 
Acknowledgments: The work is acknowledged by MOD project with code: 
2018.76.040. 
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TÓM TẮT 
MÔ HÌNH ĐẠI DIỆN CHO GIẢI THUẬT TIẾN HÓA ĐA MỤC TIÊU. 
 KHẢO SÁT VÀ VẤN ĐỀ NGHIÊN CỨU 
Bài toán tối ưu đa mục tiêu là một lớp bài toán tối ưu trong thực tế, có 
nhiều mục tiêu xung đột với nhau. Giải thuật tiến hóa tối ưu đa mục tiêu 
thường khá hiệu quả để giải các bài toán tối ưu đa mục tiêu khó. Với giải 
thuật tiến hóa đa mục tiêu, dựa trên nguyên lý quần thể, chúng ta đạt được 
một tập giải pháp tối ưu (tập giải pháp khả dụng) sau quá trình tìm kiếm. 
Chúng ta thường sử dụng quan hệ trội trong quần thể và không khó khăn để 
xác định được các giải pháp Pareto qua mỗi thế hệ. Tuy nhiên, với các bài 
toán tối ưu khó trong thực tế, cần phải có nhiều đánh giá của hàm thích nghi 
trong quá trình tìm kiếm. Để tránh các thí nghiệm tốn kém, chúng ta có thể 
dùng phương pháp mô phỏng trên máy tính để giải quyết các bài toán khó 
này. Thực tế, phương pháp này đòi hỏi chi phí tính toán và thời gian lớn cho 
quá trình mô phỏng. Vì thế, có nhiều nghiên cứu thảo luận về việc sử dụng 
mô hình đại diện cho giải thuật tiến hóa, đặc biệt là tiến hóa đa mục tiêu để 
giảm số lượng tính toán thích nghi. Bài báo này đưa ra một khảo sát ngắn về 
mô hình đại diện cho giải thuật tiến hóa tối ưu đa mục tiêu, những kết quả 
hiện tại và vấn đề nghiên cứu đặt ra. 
Từ khóa: Mô hình đại diện; Mô hình xấp xỉ; Mô hình meta; Giải thuật tiến hóa đa mục tiêu. 
Nhận bài ngày 11 tháng 01 năm 2019 
Hoàn thiện ngày 16 tháng 3 năm 2019 
Chấp nhận đăng ngày 25 tháng 3 năm 2019 
Address: 1MITI, Military Academy of Science and Technology; 
2National Defense Academy; 
3VNPT Information Technology; 
4AI Academy. 
 *Email: [email protected].