混合多群粒子群算法解决约束优化问题

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"混合多群粒子群算法在解决约束优化问题中的应用" 在实际的工程与科研领域，许多优化问题都伴随着各种类型的约束条件，这些被称为约束优化问题（COPs）。解决这类问题在优化理论中占有至关重要的地位。本文提出了一种混合多群粒子群优化（HMPSO）算法，专门针对约束优化问题进行设计。混合多群粒子群优化算法采用并行搜索操作策略，将当前的粒子群分割成若干个子群，每个子群由一个粒子群优化（PSO）算法独立驱动进行搜索。这种方法旨在增强全局搜索能力，同时保持局部搜索效率。此外，为了更有效地探索搜索空间中的潜在最优区域，该算法还结合了差异进化（DE）策略来改进每个粒子的个人最佳位置。差异进化策略通过引入变异、交叉和选择等操作，使得粒子能够跳出局部最优，寻找更优的解。在对13个标准测试函数的实验验证中，HMPSO算法的表现优于三个现有的先进方法。模拟结果表明，混合多群粒子群算法在处理约束优化问题时，具有更好的全局寻优性能和收敛速度。此研究论文展示了HMPSO在解决复杂约束优化问题上的潜力，特别是在工程设计、资源调度、机器学习模型参数优化等领域可能有广泛应用。通过结合PSO和DE的优势，HMPSO能够有效地平衡探索与开发，提高求解效率，降低陷入局部最优的风险。未来的研究可以进一步探讨如何优化子群的划分策略，以及如何动态调整DE与PSO的结合方式，以适应不同类型的约束优化问题。

40 Yong WANG, et al. A hybrid multi-swarm particle swarm optimization to solve constrained optimization problems

fect of the parameter setting. Moreover, the performance of

HMPSO is also validated on 24 benchmark test functions in

CEC2006. Finally, Section 6 concludes this paper.

       

2.1 The basic idea of PSO

As a stochastic global optimization method, PSO takes in-

spiration of the motion of a ﬂock of birds. In PSO, each po-

tential solution is regarded as a particle. A set which is com-

posed of particles is called a swarm. The movement of par-

ticles in the search space is dynamically inﬂuenced by their

personal past experience and successful experience attained

by their neighbors or the whole swarm. At the tth generation,

the position vector and the velocity vector of the ith parti-

cle in the n-dimensional search space can be represented as

x

=(x

i,1

,...,x

i,n

) and v

=(v

i,1

,...,v

i,n

). In addi-

tion, at the tth generation pbest

=(pbest

i,1

,...,pbest

i,n

)

is denoted as the best previous position of the ith particle,

and gbest

=(gbest

,...,gbest

) is denoted as the best

position of the whole swarm. During the evolution, Eq. (2)

updates each dimension of the velocity for the ith particle

for the next generation, whereas Eq. (3) updates the ith par-

ticle’s position in the search space:

t+1

i,j

= v

i,j

+ cr

(pbest

i,j

− x

i,j

)

+ cr

(gbest

− x

i,j

), (2)

t+1

i,j

= x

i,j

+ v

t+1

i,j

, (3)

where j ∈{1, 2,...,n},c is a constant known as accelera-

tion coefﬁcient, r

and r

are two separately generated uni-

formly distributed random numbers in the range [0,1].

It is necessary to note that in terms of the above version

of PSO, since it lacks the mechanism to control the magni-

tude of the velocities, the velocities and position of the par-

ticles might careen toward inﬁnity, which results in a kind

of explosion and divergence [17]. To overcome this de-

fect, the velocities are usually conﬁned within the range of

[−



max



max

Shi and Eberhart [18] addressed the aforementioned ex-

plosion problem by further incorporating a weight parame-

ter into the previous velocity of the particle. As a result, the

velocity and position updates are:

t+1

i,j

= wv

i,j

+ c

(pbest

i,j

− x

i,j

)

+ c

(gbest

− x

i,j

), (4)

t+1

i,j

= x

i,j

+ v

t+1

i,j

, (5)

where w denotes the inertia weight, c

and c

are the ac-

celeration constants, r

and r

are two separately generated

uniformly distributed random numbers in the range [0,1].

Clerc and Kennedy [19] proposed an alternative version

of PSO in which the velocity adjustment is given as follows:

t+1

i,j

= χ(v

i,j

+ c

(pbest

i,j

− x

i,j

)

+ c

(gbest

− x

i,j

)), (6)

where

χ =

|2 − φ −



− 4φ|

, (7)

and φ = c

+ c

, φ>4. By making use of the constric-

tion coefﬁcient χ, the above version of PSO eliminates the

parameter



max

By analyzing Eq. (6), Krohling and Coelho [10] con-

cluded that the mean values of the two stochastic coefﬁ-

cients for the local term (pbest

i,j

− x

i,j

) and the global term

(gbest

−x

i,j

) lies in the interval [0.72, 0.86]. Consequently,

a proper choice for the probability distribution that generates

stochastic coefﬁcients is the absolute value of the Gaussian

probability distribution with zero mean and unit variance,

i.e., abs(N(0, 1)) and, as a result, the velocity equation is

updated according to:

t+1

i,j

= |randn|(pbest

i,j

− x

i,j

)

+ |Randn|(gbest

− x

i,j

), (8)

where |randn| and |Randn| are positive random numbers

generated using abs(N(0, 1)).

2.2 The basic idea of DE

Differential evolution (DE) was proposed by Storn and Price

[20] in 1995 which is a stochastic and population-based op-

timization algorithm, and uses vector differences to perturb

the vector population.

Initially, DE starts with a population consisting of Nn-

dimensional vectors which have the form:

x

= {x

i,1

i,2

,...,x

i,n

},i=1, 2,...,N, (9)

where t denotes the generation number. These vectors are

randomly selected on the intervals: [L

], i =1, 2,...,n.

During the evolution, each of the N vectors undergoes mu-

tation, crossover and selection operations.

There are several versions for DE. Next, the most popular

version of DE called “DE/rand/1/bin” will be introduced.

Mutation operation: For each given vector x

at genera-

tion t, three vectors x

, x

,andx

are randomly selected

such that the indices i, r

, r

and r

are distinct. x

is called

the target vector. Afterward, the weighted difference of two

of the vectors is added to the third to form a mutant vector

v

= {v

i,1

i,2

,...,v

i,n

v

= x

+ F (x

− x

). (10)

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