CN110751258A - Hyper-sphere search algorithm based on differential evolution - Google Patents

Hyper-sphere search algorithm based on differential evolution Download PDF

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CN110751258A
CN110751258A CN201911023202.7A CN201911023202A CN110751258A CN 110751258 A CN110751258 A CN 110751258A CN 201911023202 A CN201911023202 A CN 201911023202A CN 110751258 A CN110751258 A CN 110751258A
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hypersphere
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赵海波
齐玉文
李恩龙
王艳秋
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CRRC Changchun Railway Vehicles Co Ltd
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Abstract

A hypersphere search algorithm based on differential evolution defines an initial population number NpopCenter number of hypersphere NSCUpper and lower limits r of radius of hypersphere searchmin，rmaxProbability of change of superball motion angle PrangleThe method comprises 6 steps of initializing an algorithm, moving a result to the optimal, carrying out differential evolution, recovering virtual particles, re-determining a hyper-sphere center and judging convergence, wherein the 6 steps are compact in flow connection, the overall algorithm design is simple, the flow is clear, and the practicability is high. The algorithm can improve the global search capability and enhance the accuracy of the obtained solution under the condition of not losing the optimal solution, and provides a new optimization technology for the field of rail transit.

Description

Hyper-sphere search algorithm based on differential evolution
Technical Field
The invention relates to the field of meta-heuristic intelligent optimization, in particular to a hypersphere search algorithm based on differential evolution.
Background
Optimization problems have always been a non-avoidable class of problems in engineering practice and scientific research. The solution of the optimization problem mainly comprises two methods, namely an analytic method and an intelligent optimization algorithm. The analytic method generally needs a mathematical model to be capable of corresponding to the existing mathematical calculation method, and simplifies the problem which cannot be solved by the existing mathematical analysis optimization algorithm, otherwise, the analytic method cannot be used.
Disclosure of Invention
The invention aims to provide a hypersphere search algorithm based on differential evolution, which can improve the global search capability and enhance the accuracy of the obtained solution under the condition of not losing the optimal solution.
In order to achieve the purpose, the invention provides a hypersphere search algorithm based on differential evolution, which is characterized by adopting the following technical scheme:
defining an initial population number NpopCenter number of hypersphere NSCUpper and lower limits r of radius of hypersphere searchmin，rmaxProbability of change of superball motion angle PrangleThe differential evolution cross probability MR and the differential evolution mutation probability CR comprise the following main steps:
1. initializing particles, namely defining initial population number, hypersphere center number, hypersphere search radius upper and lower limits, hypersphere motion angle change probability and differential evolution intersection/variation probability, calculating objective function values of all particles, and preliminarily determining hypersphere center SCs and residual particle membership;
2. moving the particles to the corresponding SC, searching the particles, determining whether the objective function value of any particle is lower than the SC of the particle, if so, exchanging the SC by the particles, otherwise, entering the step 3;
3. differential evolution, namely determining whether differential variation is carried out or not, and if so, carrying out differential variation; otherwise, determining whether to perform differential crossing, and if so, performing differential crossing on each dimension; otherwise, comparing the target function update particles and reselecting each SC;
4. calculating a total objective function, selecting and recycling virtual particles, and searching new SC (single carrier) for the particles based on the distribution probability of the hyper-sphere center; searching whether an SC without particles exists, if so, changing the SC without particles into particles, otherwise, entering the step 5;
5. selecting a new SC and its particles, namely, at the end of each iteration, sequencing all particles and SCs according to objective function values thereof, and selecting the best particles as the new SCs of the next iteration;
6. and in the process of reaching the maximum iteration number 1000, whether the difference between the best SCs among algebras is lower than a preset threshold value is judged, if yes, the process is ended, and if not, the process returns to the step2.
The positive effects produced by the scheme adopted by the invention are as follows:
the six-step flow in the invention is closely connected, the whole algorithm design is simplified, the flow is clear, and the practicability is strong. The algorithm is initialized in the first step, the result moves to the optimal state in the second step, and aiming at the problem of weak algorithm global search capability in the optimization field, the third step in the method is a differential evolution step, so that the global search capability of the algorithm is improved, more reasonable results can be obtained by solution, the global optimization capability of the algorithm can be enhanced under the condition that the optimal solution is not lost, the accuracy of algorithm solving is enhanced, the optimization in the fourth step and the fifth step is more refined, whether the algorithm is finished or not is judged in the sixth step, and the optimal decision is finally obtained. The algorithm provides a new optimization technology for the field of rail transit.
Drawings
Fig. 1 shows a schematic flow diagram in the present invention.
Detailed Description
As shown in fig. 1, a hypersphere search algorithm based on differential evolution includes the following steps:
defining an initial population number NpopCenter number of hypersphere NSCUpper and lower limits r of radius of hypersphere searchmin，rmaxProbability of change of superball motion angle PrangleDifferential evolution cross probability MR, differential evolution mutation probability CR. Taking the example of finding the minimum value in the search space, the process of introducing the algorithm includes the following main steps:
step1 initialization
Step1.1 defines the initial population number NpopCenter number of hypersphere NSCUpper and lower limits r of radius of hypersphere searchmax，rminProbability of change of superball motion angle PrangleDifferential evolution cross probability MR, differential evolution mutation probability CR;
step1.2 hyper-sphere search algorithm based on differential evolution from an initial set of NpopSolutions were started, which were randomly generated. Each decision variable xiWith uniform probability from [ Xi,min,Xi,max]Is randomly selected. Each solution is called a particle, and an objective function is calculated for each particle;
step1.3 classifies particles in ascending order based on their objective function values, and selects the best NSCParticles (from the top of the list) as hyper-Sphere Centers (SCs); in the N-dimensional optimization problem, the particles are represented by a 1 XN vector [ p ]1,p2,...,pN]And (4) showing. p is a radical ofiN is a decision variable. By evaluation (p)1,p2,...,pN) An objective function of f, i.e. f (p)1,p2,...,pN) To determine an objective function value for the particle;
Step1.4 Npopafter the particles are generated, the top N of the rank is selectedSCThe particles of (2) are referred to as SCs. Considering the advantages of SCs, the remaining particles are distributed in SCs in inverse proportion to their objective function values. To scale the particles, the Objective Function Difference (OFD) for each SC is defined as the difference between the objective function value of that SC and the maximum objective function value of the SCs. In other words, OFDSC＝fSC-maxSCs{ f }. Thus, the normalized dominance of each SC is defined by the following equation:
Figure BDA0002247877960000031
then, the initial number of particles belonging to the SC will equal round { DSC×(Npop-NSC) Which is randomly selected by each SC from the remaining particles.
Step2 search
The step2.1 particle seeks a better solution by searching the sphere-constrained space with a predefined center. The radius of the sphere is r, the distance between the particle and the center. For this purpose, the origin of coordinates is set at the center of the sphere. The search program will search by changing the particle parameters in the spherical coordinates: radius and angle, i.e., r and θ;
step2.2 has N-1 angles (θ s) in the N-dimensional problem, where any one angle change will result in the particle moving in the search space in the hyper-sphere search algorithm based on differential evolution, each angle of the particle changes at α radians, with a probability of Pr change for each radianangleRandomly select α in each iteration between (0,2 π) with a uniform distribution;
step2.3 after changing all angles of the particle, the distance between the particle and the center is [ r ]min,rmax]Randomly selected from the above. In an N-dimensional hypersphere, r is calculated as follows:
Figure BDA0002247877960000041
after changing thetas and r and evaluating f, the searching process of the particle in the searching space is completed;
after step2.4 searches for a particle in its sphere, the particle may reach a position with a lower value of the objective function than its corresponding SC. In this case, the SC and the particle's label are exchanged, i.e. the original SC becomes the particle of the new SC.
Step3 differential evolution
Step3.1 for each particle belonging to SC, if rand () > MR (MR is the probability of variation)
P＝p1+F(p3-p2)+rand(SCbest-p)
Wherein p is1、p2、p3Is 3 particles selected randomly; f is a scaling factor; SC (Single chip computer)bestIs the best hyper-sphere center, i.e. the global optimum; rand is a random value between 0 and 1; p is a mutated particle; MR is mutation probability, and proper MR value can improve the calculation efficiency of the algorithm.
Otherwise, judging whether to perform differential crossing (CR is crossing probability)
Figure BDA0002247877960000042
Step3.3 calculating the target function of the new particle D, and updating the particle if the target function is smaller than the original particle;
step3.4 reselects the SCs after the target function is compared to update the particles.
Step4 virtual particle recovery
Particles that are searched in an inappropriate space are referred to as dummy particles.
The Step4.1 particle set should be classified according to their Set Objective Function (SOF) values to find the worst set with virtual particles. The SOF of the set is mainly influenced by the objective function value of the SC, which is not so important for the particles. The problem is modeled by defining the SOF for each set by the following equation:
SOF＝fSC+γmean{fparticles of SC}
a small value of γ allows a set of SOF to be determined from the objective function values of the SC, and increasing γ will increase the effect of the particles in determining the SOF;
the process of Step4.2 virtual particle reclamation is modeled by selecting some (e.g., one) virtual particles from the hypersphere with the largest SOF and assigning them (it) to other SCs. To this end, the difference in SOF (DSOF) for each set is simply determined by the following equation:
Figure BDA0002247877960000051
assigning the particle to one of the SCs using the calculated DSOF;
step4.3 calculates the distribution probability (AP) for each SC, and the equation is as follows:
Figure BDA0002247877960000052
forming a vector
Figure BDA0002247877960000053
To divide the particles in SCs based on their APs, i.e. virtual particles with APsiIs assigned to the ith SC. In short, the worst group (with the highest SOF) will lose its ghostingPseudo-particles. The particle looks for a new SC among all SCs based on its AP. It should be noted that if an SC has no particles, then this SC will change to particles and use this process to find a new SC.
Step5 re-centering the hyper-sphere
At the end of each iteration, all particles and SCs are sorted according to their objective function values, and the best particle is selected as the new SCs for the next iteration.
Step6 judges convergence
The algorithm is terminated if it reaches one of the following conditions:
(a) a maximum number of iterations is reached (e.g., the maximum number of iterations equals 1,000).
(b) The difference between the best SCs over a certain number of generations in the iteration becomes lower than a preset threshold.

Claims (1)

1. A hypersphere search algorithm based on differential evolution is characterized in that the following technical scheme is adopted:
defining an initial population number NpopCenter number of hypersphere NSCUpper and lower limits r of radius of hypersphere searchmin，rmaxProbability of change of superball motion angle PrangleThe differential evolution cross probability MR and the differential evolution mutation probability CR comprise the following main steps:
(1) initializing particles, namely defining initial population number, hypersphere center number, hypersphere search radius upper and lower limits, hypersphere motion angle change probability and differential evolution intersection/variation probability, calculating objective function values of all particles, and preliminarily determining hypersphere center SCs and residual particle membership;
(2) moving the particles to the corresponding SC, searching the particles, determining whether the objective function value of any particle is lower than the SC of the particle, if so, exchanging the SC by the particles, otherwise, entering the step 3;
(3) differential evolution, namely determining whether differential variation is carried out or not, and if so, carrying out differential variation; otherwise, determining whether to perform differential crossing, and if so, performing differential crossing on each dimension; otherwise, comparing the target function update particles and reselecting each SC;
(4) calculating a total objective function, selecting and recycling virtual particles, and searching new SC (single carrier) for the particles based on the distribution probability of the hyper-sphere center; searching whether an SC without particles exists, if so, changing the SC without particles into particles, otherwise, entering the step 5;
(5) selecting a new SC and its particles, namely, at the end of each iteration, sequencing all particles and SCs according to objective function values thereof, and selecting the best particles as the new SCs of the next iteration;
(6) and in the process of reaching the maximum iteration number 1000, whether the difference between the best SCs among algebras is lower than a preset threshold value is judged, if yes, the process is ended, and if not, the process returns to the step2.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114967428A (en) * 2022-07-29 2022-08-30 济南大学 Unmanned life buoy optimal robust control method based on improved particle swarm algorithm

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114967428A (en) * 2022-07-29 2022-08-30 济南大学 Unmanned life buoy optimal robust control method based on improved particle swarm algorithm
CN114967428B (en) * 2022-07-29 2022-11-22 济南大学 Unmanned lifebuoy optimal robust control method based on improved particle swarm algorithm

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