2017 9th International Conference on Intelligent Human-Machine Systems and Cybernetics
Adaptive Constrained Multi-objective Biogeography-based Optimization Based on Two-stage Elite Selection Jue Wang, Bo Li, Yinghong Cao College of Information Science and Engineering Dalian Polytechnic University Dalian, China e-mail:
[email protected],
[email protected],
[email protected] optimization based on two-stage elite selection (ACMBBO) is proposed, which the constrained multi-objective optimization model for BBO is established and the relevant evolution strategy for BBO algorithms is improved. The remainder of this paper is organized as follows. Section II presents a detailed description of ACMBBO. Section III reports on the simulation results, and shows the performance of ACMBBO in comparison with other well-known optimization algorithms. Section IV presents the study conclusions.
Abstract— a new adaptive constrained multi-objective biogeography-based optimization based on two-stage elite selection, ACMBBO, is proposed to solve constrained multiobjective optimization problems. According to the feature on constrained multi-objective and the evolutionary mechanism of BBO, the model of constrained multi-objective optimization which applies to BBO is built. In the model, the habitat suitability index, which combines with the degree of feasible and the Pareto dominance relation between the individuals, is redefined. Moreover, a new mechanism based on two-stage elite selection is set to preserve the elitist of population individuals. Also dynamic migration strategy is designed to improve the ability for exploitation and the utilization of the better individual. Numerical experiments have shown that ACMBBO is competitive with current other constrained multiobjective optimization algorithms on the convergence and the distribution, and is capable of solving the complex CMOPs more effectively and efficiently.
II.
In view of the fact that BBO algorithm itself does not have the ability to deal with constrained multi-objective optimization problems, a multi- objective optimization model need to be designed in this paper, which is suitable for BBO based on İ constraint method. In addition, in order to improve the constrained optimization ability of multiobjective optimization, the efficiency of BBO algorithm must be ensured.
Keywords-constrained multi-objective optimization; two-stage elite selection; biogeography-based optimization; dynamic migration strategy
I.
A. Modified determination method of HSI In the BBO algorithm, habitat suitability index (HSI) is an indicator of measuring the habitat quality. For CMOPs, in consideration of the characteristics of multi-objectives, it is necessary to determine HSI by combining Pareto dominance among habitat individuals. However, it is far from enough to define the HSI which is not enough to evaluate the quality of the habitat in this way and the constraints should also be taken into account fully. Thus it is ideal to design a determination mechanism of HSI, which can take into account the Pareto domination relationship and the degree of constraint satisfaction of individual itself. For this purpose, this paper proposes a new determination method of habitat suitability index HSI on the basis of the idea of İ constraint domination mentioned above. First, the individuals in habitat population H={xi,i=1,2,…,NP} is ranked in ascending order in accordance with the adaptive İ weighting constraint violation, and form a new population H. Then, the İ feasible habitat population Hİfea={xi| φ ( xi ) =0} of current new population H is determined and the non-dominance degree Fi of individual xi in Hİfea is calculated by formula (1).
INTRODUCTION
Constrained multi-objective optimization problems (CMOPs)[1] can be found in many real-world applications [2,3]. Due to the conflicts of objectives, the interference among constraints and the interrelationship between the constraints and the objectives, there are some difficulties in solving CMOPs. Many optimization methods have shown a certain search performance, but there are still problems that it is easy to fall into the local optimal frontier and cannot take into account the convergence and distribution. Thus, the key to solving the CMOPs lies in the effectiveness of handling constraints, the efficiency of evolutionary mechanism, and the balance between the constraint handling technology and multi-objective evolutionary strategy. In order to obtain better solution performance on CMOPs, this paper chooses İ constraint method[4] as constraint handling strategy. On the basis of biogeography-based optimization algorithm (BBO)[5] with superior performance in solving global optimization problems, combined the characteristics of CMOPs, a new adaptive constrained multi-objective biogeography-based 978-1-5386-3022-8/17 $31.00 © 2017 IEEE DOI 10.1109/IHMSC.2017.193
THE PROPOSED ALGORITHM
365
Fi =
§ 1 k x j , xk ∈ H ε fea ∧ x j ; xk , ¨ xi ∈H ε fea , x j ; xi © NP
¦
{
dominated habitats, but also to the maintenance of the diversity of infeasible individuals. The infeasible individuals close to the isolated feasible region may be effectively preserved. The pseudo code for the individual reservation method is as Figure 2.
(1)
} ) , i, j, k ∈{1, 2,..., NP}
H ε fea = { xi φ ( xi ) = 0}
1 for i = 1 to N f
Where xi, xj, xk ∈ H is the habitat individual including ndimensional suitability index variables (SIV). Finally, the HSI Gi of the individual xi in the new habitat population H is determined as formula (2).
Gi = max(φ ( xi ), 0) + Fi ,
2
dij = Euclidean_distance( xi , x j );
3
4 endfor 5 endfor 6 while size(H f ) >N k
'
Fi , φ ( xi ) = 0, Fi ' = ® ¯ Fmax , φ ( xi ) ≠ 0.
for j = 1 to N f − 1
(2)
7
select a pair of individuals x A , xB of the smallest dij ;
8
find out respectively the second smallest value l A and lB of d Aj , d Bj ;
B. Two-stage elite selection method After determining the HSI of the individuals in population H, we need to retain the NP better habitat individual by the HSI as the current optimal population Hn to participate in further evolution. In general, if the number of non-dominated habitat np in H is less than NP, then the NP better individuals should be the non-dominated individuals and the individuals with lower degree of domination; On the contrary, the NP individuals with the distribution of more uniform should be reserved as Hn. Aiming at the shortcomings of the distributed maintenance methods of CMOPs, a competitive selection mechanism based on dynamic distance matrix is designed to maintain the distribution of population. First, the distance matrix is established among the habitat individuals. Second, a pair of individual with higher degree of aggregation is selected to compete, and the individuals of highest aggregation are eliminated. Finally, the matrix is updated dynamically. The pseudo code of this competitive selection mechanism is shown as Figure 1. Where Hf={xi, i=1,2,...,Nf} is the current habitat population without competing and selecting; Nf is the number of the habitat individuals without competing and selecting; Hk={xi, i=1,2,...,Nk} is the habitat population that have not been preserved; Nk is the number of the habitat individuals that have not been preserved. In CMOPs, due to the constraint conditions, the feasible region can be divided into several isolated regions. In this case, the algorithm is easy to fall into one or several sub regions, and cannot search the entire feasible region. Therefore, for CMOPs, we not only need to ensure the distribution of non-dominated habitat individuals in the process of evolution, but also in the early stages of evolution, it is very necessary for infeasible individuals in population to maintain the diversity. Based on the above ideas, this paper proposes a two-stage elite reservation method for BBO, which the distribution maintenance mechanism based on the dynamic distance matrix is not only used to maintain the distribution of individuals in non-
9
if l A
