WO2022007376A1 - 基于贝叶斯自适应共振的多目标多模态粒子群优化方法 - Google Patents
基于贝叶斯自适应共振的多目标多模态粒子群优化方法 Download PDFInfo
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- G06F30/15—Vehicle, aircraft or watercraft design
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- the invention relates to the technical field of optimization algorithms, in particular to a multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance.
- Some cluster-based particle swarm optimization algorithms often need to pre-set the number of clusters according to the number of solutions, but in reality it is difficult to know in advance how many solutions there are for multi-objective multi-modal functions, so the past methods generally take different Experiments are carried out with the number of clusters, and the optimal number of clusters is selected according to the experimental results, which brings great uncertainty and problem dependence to the optimization problem.
- the present invention provides a multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance, the purpose of which is to obtain as many target spaces as possible in order to better solve the multi-objective multi-modal optimization problem
- the Pareto frontier and the Pareto optimal solution set of the decision space ensure the diversity of solutions in the target space and decision space.
- the multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance includes the following steps:
- Sort the particles of each population according to the non-dominated sorting method and the special crowding distance according to the non-dominated sorting method, hierarchically sort all the particles in the various groups in the target space, and sort all the particles in the first non-dominated layer after sorting.
- the particles of the layer are sorted in descending order according to the size of the special crowding distance;
- the particles of each population are evolved based on the global optimal particle swarm optimization algorithm
- the decision space of the particle is the space composed of the value range of each dimension of the particle; the target space is the space composed of the value range of the objective function determined by the decision space.
- Non-dominated in S2 means: in a multi-objective problem, if one solution A is better than another solution B in all objectives, then solution A dominates solution B; if one solution A is not dominated by other solutions, Then the solution A is called a non-dominated solution.
- the crowding distance is an index of the crowding degree between the target solution of a particle and the target solutions of its adjacent particles in the target space.
- the specific content of S1 includes:
- the calculation method of the posterior probability of the particle for the existing population is:
- K represents the number of population existing
- y l j represents the populations
- the estimated prior probability of the jth population represents the possibility of population j to particle x;
- ⁇ j represent the estimated mean vector and covariance matrix of the j population
- the population sample size is represented by the hypervolume S J , which is the determinant of the Gaussian covariance matrix.
- the hypervolume is the product of variances in each dimension:
- d represents the particle dimension, represents variance
- M J represents the number of particles in the J population
- I represents the identity matrix
- the non-dominated sorting method includes the following:
- the target solution of these particles is divided into the second non-dominated layer in the target space;
- step (2) all particles in the population are sorted by layers.
- the calculation method of the special crowding distance is as follows:
- M represents the number of all objective functions
- CD im, obj represent the crowding distance of particle i in the dimension of objective function f m (x)
- the calculation method is:
- f m (x i+1 ) and f m (x i-1 ) represent the adjacent target solutions of the objective function solution f m ( xi ) of particle i, and f m (x max ) and f m (x min ) Represents the maximum and minimum values of the objective function f m (x);
- CD i, obj ′ represents the crowded distance of particle i in the target space after normalization
- CD i, obj ′ represents the crowded distance of particle i in the decision space after normalization
- M represents the dimension of the target space
- N represents the decision space dimension
- CD avg, obj ′ and CD avg, dec ′ represent the normalized average crowded distance in the target space and the average crowded distance in the decision space, respectively.
- V′ i wV i +c 1 r 1 (Pbest i -X i )+c 2 r 2 (Gbest-X i )
- X i is the current particle position
- X i ' is the updated particle position
- V i is the current particle velocity
- V i ' is the updated particle velocity
- Pbest is the individual best of each particle
- Gbest is The population is globally best, it is subGbest;
- each non-dominated solution set of each population are connected end to end to form a ring topology, and each non-dominated solution set and its adjacent two non-dominated solution sets are regarded as a neighborhood;
- the first particle in the neighborhood file is regarded as the best Nbest of the neighborhood, and then the global best subGbest of each population is updated.
- the update method is:
- V′ i wV i +c 1 r 1 (Pbest i -X i )+c 2 r 2 (Nbest i -X i )
- the termination condition is generally a specified number of iterations or the obtained solution satisfies a certain error range.
- the specific steps of S6 are: after the iteration is completed, the non-dominated particles in each non-dominated solution set are Pareto optimal solutions in the decision space, and each Pareto optimal solution is substituted into the objective function to obtain Pareto frontier in target space.
- the following steps are further included: determining relevant parameters of the particle swarm, randomly generating N particles in the decision space, and initializing the particle swarm.
- the relevant parameters include the total number of particles N, the particle dimension D, the inertia weight w, the speed control parameter c1 and the speed control parameter c2; the speed and position of each particle are initialized.
- the present invention provides a multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance, which has the following advantages compared with the prior art:
- the present invention uses Bayesian adaptive resonance theory to divide the particle swarm. Unlike the previous clustering methods based on Euclidean distance and k-means, this method does not need to pre-set the number of clusters, and uses The particles are clustered unsupervised and adaptively with respect to the posterior probability of all existing clusters;
- the present invention is based on a multi-swarm mechanism, and different sub-populations simultaneously independently track and save the Pareto optimal solution set of the decision space, which is beneficial to the solution of multi-modal problems; and the present invention also adopts non-dominated sorting and special crowding.
- the distance sorts the particles of various groups, and uses the global optimal particle swarm algorithm to independently mine each sub-population, which is conducive to discovering the distribution of non-dominated solution sets and Pareto frontiers, which is more conducive to multi-objective multi-mode solution of state problems;
- each population can obtain guiding information from the adjacent population, so as to realize the communication between the populations. It is beneficial to maintain the diversity of particles, and use the local search based on ring topology to further improve the search efficiency and improve the space exploration ability of the algorithm;
- Pareto optimal solution set of the decision space and the corresponding Pareto optimal solution of the target space can be output at the same time, and the solutions of multi-objective and multi-modal problems can be presented more intuitively.
- FIG. 1 accompanying drawing is the overall step flow chart of the multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance provided by the present invention
- Fig. 2 is the algorithm flow chart of the multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance provided by the present invention
- FIG. 3 is a schematic diagram of a decision space and a target space in the multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance provided by the present invention
- FIG. 5 is a schematic diagram of a ring topology formed by a non-dominated solution set in the multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance provided by the present invention.
- the embodiment of the present invention discloses a multi-objective multi-modal particle swarm optimization method based on Bayesian adaptive resonance.
- Step 1 Determine the relevant parameters of the particle swarm and initialize the particle swarm.
- the relevant parameters here mainly include: the total number of particles N, the particle dimension D, the inertia weight w, and the learning factors c1 and c2.
- the total number of particles N is crucial to the performance of the algorithm. If the number of particles is too small, the diversity of the population will be insufficient, and it will not be enough to fully cover the decision space, resulting in poor decision performance. Conversely, if the population size is too large, it will consume too much computing resources, so it is necessary to choose an appropriate total number of particles. This parameter needs to be set based on actual problem requirements, prior experience, and some preliminary experiments.
- the particle dimension D is determined by the number of variables. In general, the particle dimension is equal to the number of decision variables.
- the inertia weight w is the ability of the particle to maintain the motion state of the previous moment.
- the learning factors c1 and c2 are mainly used to control the degree of influence of the particle by individual cognition and social cognition. Usually, the value of w is 0.7298, and the values of c1 and c2 are both 2.05.
- the stability and controllability of the aircraft are two very important characteristics of the aircraft. If the stability is good, the force and torque of the aircraft to resist the change of the flight state will be large, the response of the aircraft to the pilot's control will be slow, and the controllability of the aircraft will be reduced. Correspondingly worse. How to reconcile the relationship between the stability and maneuverability of an aircraft is a very worthy trade-off for modern aircraft design. In order to balance the longitudinal stability and maneuverability of the aircraft, it is necessary to determine the position of the aircraft's focus and center of gravity in the aircraft design stage. The focus and center of gravity of the aircraft are two important factors that affect the stability and maneuverability of the aircraft.
- the stability and maneuverability of the aircraft are regarded as objective functions f 1 , f 2
- the focus and the position of the center of gravity of the aircraft are regarded as decision variables x 1 , x 2
- there are 2 decision variables so the The particle dimension N determined in the step is 2.
- the decision space of the particle is shown on the left side of Figure 3, where "x 1 , x 2 " is the decision variable; it is the space composed of the value range of each dimension variable of the particle; the target space is shown on the right side of Figure 3 , is the space composed of the value range of the objective function determined by the decision space, where "y 1 , y 2 " is the objective function.
- Randomly generate N particles in the decision space and initialize the speed and position of each particle.
- N particles are randomly generated within the range of the focus and the center of gravity of the aircraft, and the abscissa x 1 of the particles represents the size of the aircraft.
- the focus position, the ordinate x 2 of the particle represents the position of the particle's center of gravity.
- Step 2 Use Bayesian adaptive resonance theory to divide the particle swarm into several populations.
- Bayesian adaptive resonance is regarded as a clustering method here, and the process is shown in Figure 4.
- the specific method is as follows:
- the calculation method of the posterior probability of the particle for the existing cluster is:
- K represents the number of population existing
- y l j represents the populations
- the estimated prior probability of the jth population represents the possibility of population j to particle x;
- ⁇ j represent the estimated mean vector and covariance matrix of the j population
- the population sample size is represented by the hypervolume S J , which is the determinant of the Gaussian covariance matrix.
- the hypervolume is the product of variances in each dimension:
- d represents the particle dimension, represents variance
- M J represents the number of particles in the J population
- I represents the identity matrix
- N particles can be divided into k populations.
- Step 3 Sort the particles of each population according to the non-dominated sorting method and the special crowding distance.
- solution A dominates solution B if one solution A is better than another solution B on all objectives.
- a solution A is called a non-dominated solution if it is not dominated by other solutions.
- the target solution of these particles is divided into the first non-dominated layer in the target space.
- the target solution of these particles is divided into the second non-dominated layer in the target space.
- All particles in the population are sorted hierarchically according to the above method.
- the crowding distance is an index of the crowding degree between the target solution of a particle and the target solutions of its adjacent particles in the target space.
- M represents the number of all objective functions
- CD im, obj represent the crowding distance of particle i in the dimension of objective function f m (x)
- the calculation method is:
- f m (x i+1 ) and f m (x i-1 ) represent the adjacent target solutions of the objective function solution f m ( xi ) of particle i
- f m (x max ) and f m (x min ) represents the maximum and minimum values of the objective function f m (x).
- the multi-objective multi-modal problem needs to consider not only the diversity of the target space, but also the diversity of the decision space. Therefore, if the crowded distance of the target space is extended to the decision space, the crowded distance of the decision space CD i can be obtained in the same way, dec ; replace the objective function in the above method with a decision function to obtain the decision space crowding distance CD i, dec ;
- CD i, obj ′ represents the crowded distance of particle i in the target space after normalization
- CD i, obj ′ represents the crowded distance of particle i in the decision space after normalization
- M represents the dimension of the target space
- N represents the decision space dimension
- CD avg, obj ′ and CD avg′dec ′ represent the normalized average crowded distance of the target space and the average crowded distance of the decision space, respectively.
- the non-dominated particles in the first non-dominated layer are sorted in descending order according to the size of the special crowding distance, and the first non-dominated particle has the largest special crowding distance.
- Step 4 Save the sorted non-dominated solutions of each population in the set of non-dominated solutions of various populations.
- the non-dominated solution set of each population is established, and the non-dominated particles sorted in descending order according to the special crowding distance are stored in the non-dominated solution set. If a new non-dominated solution is generated subsequently, it will be added to the non-dominated solution set, and the particles dominated by the new non-dominated solution will be deleted.
- Step 5 The particles of each population are evolved using the global optimal particle swarm optimization algorithm, and the method is as follows:
- V′ i wV i +c 1 r 1 (Pbest i -X i )+c 2 r 2 (Gbest-X i )
- X i is the current particle position
- X i ' is the updated particle position
- V i is the current particle velocity
- V i ' is the updated particle velocity
- Pbest is the individual best of each particle
- Gbest is The population is globally best, it is subGbest;
- Step 6 Use the particle swarm optimization algorithm based on ring topology for local search, the method is as follows:
- each solid circle represents a non-dominated solution set of a population, where "Nset" is a non-dominated solution set.
- the three non-dominated solution sets included in each dashed line are a neighborhood.
- V′ i wV i +c 1 r 1 (Pbest i -X i )+c 2 r 2 (Nbesti -X i )
- Step 7 Repeat the evolution of various groups and the local search process based on ring topology until the termination condition is satisfied.
- the new global best subGbest obtained in step 6 update the particle velocity and position in the population according to the rules in step 5, and then update the global best according to the rules in step 6, and iterate continuously until the termination condition is satisfied.
- it is a specified number of iterations or the obtained solution satisfies a certain error range.
- Step 8 Output the non-dominated particles in each non-dominated solution set and their corresponding Pareto fronts.
- the non-dominated particles in the non-dominated solution set of each population are the Pareto optimal solutions in the decision space, and they can be brought into the objective function to obtain the Pareto frontier in the objective space.
- the final output Pareto front is the optimal combination of aircraft stability and maneuverability, and the non-dominated particles in the non-dominated solution set are corresponding to the optimal combination of aircraft stability and maneuverability.
- the invention can complete the construction of the multi-objective multi-modal particle swarm method based on Bayesian adaptive resonance.
- the clustering method based on Bayesian adaptive theory is used to divide the particle swarm into multiple sub-populations in the decision space, and then In each population, sort each particle in various groups according to the non-dominated sorting method and the special crowding distance, and store the non-dominated solution into the non-dominated solution set, and the first one is regarded as the global optimum of the population; Then, the particles in the population are updated using the individual optimality of the particles and the global optimality of the population; the non-dominated solution sets are connected end to end to form a closed ring topology, and the particle swarm optimization algorithm based on ring topology is used for local exploration; repeat the above Two update and exploration processes are performed until the termination condition is met, outputting all non-dominated solution sets and Pareto fronts.
- Using this technology is beneficial to solve the optimization problem of multi-objective and multi-modal functions. It can help people to plan multiple alternative schemes to achieve multiple objectives in reality, provide redundant backup methods, and improve the reliability of engineering practice activities.
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Claims (10)
- 基于贝叶斯自适应共振的多目标多模态粒子群优化方法,其特征在于,包括以下步骤:S1.通过贝叶斯自适应共振理论将粒子群划分为若干个种群;S2.根据非支配排序法和特殊拥挤距离对各个种群的粒子进行排序:根据非支配排序法在目标空间中对各种群中的所有粒子进行分层排序,对排序后位于第一层非支配层的粒子按照特殊拥挤距离的大小进行降序排序;S3.将排序后的各个种群的非支配解保存在各种群的非支配解集中;S4.各个种群的粒子利用基于全局最优粒子群优化算法进化;S5.利用基于环形拓扑的粒子群优化算法进行局部搜索;S6.重复各种群的进化和基于环形拓扑的局部搜索过程直至满足终止条件;S7.输出各个非支配解集中的非支配粒子及相对应的帕累托前沿。
- 根据权利要求1所述的基于贝叶斯自适应共振的多目标多模态粒子群优化方法,其特征在于,S1的具体内容包括:S11.种群选择:为了将不同粒子划分到合适的种群中,计算每个粒子相对所有已存在种群的后验概率最大值,最大后验概率对应的种群作为获胜种群;粒子对于已存在种群的后验概率的计算方法为:根据后验概率最大的种群选择获胜种群j:S12.匹配测试:如果获胜种群的样本点容量小于警戒阈值S MAX,则执行S13;否则,根据S11寻找下一个获胜种群;如果所有种群的样本点容量都大于警戒阈值S MAX,则建立新的种群,新种群的均值向量为粒子本身;种群样本容量使用超体积S J表示,是高斯协方差矩阵的行列式,对于对角协方差矩阵,超体积为每一维的方差乘积:S13.学习更新:根据新粒子调整获胜种群的均值向量和协方差矩阵:
- 根据权利要求1所述的基于贝叶斯自适应共振的多目标多模态粒子群优化方法,其特征在于,非支配排序法包括以下内容:(1)将粒子代入到多个目标函数中,得到粒子所代表的目标解;(2)若种群中粒子没有被其他粒子所支配,则在目标空间中将这些粒子的目标解划分为第一层非支配层;若种群中的粒子除了被第一层非支配层的粒子支配以外,没有被其他粒子所支配,则在目标空间中将这些粒子的目标解划分为第二层非支配层;按步骤(2)的方法依次对种群中的所有粒子进行分层排序。
- 根据权利要求1所述的基于贝叶斯自适应共振的多目标多模态粒子群优化方法,其特征在于,特殊拥挤距离的计算方法如下:(1)计算目标空间拥挤距离CD i,obj:其中,M表示所有目标函数的个数,CD im,obj表示粒子i在目标函数f m(x)维度的拥挤距离,计算方法为:其中,f m(x i+1)和f m(x i-1)表示粒子i的目标函数解f m(x i)的临近目标解,f m(x max)和f m(x min)表示目标函数f m(x)的最大值和最小值;同理求得决策空间拥挤距离CD i,dec;(2)对目标空间和决策空间的拥挤距离进行归一化处理;其中,CD i,obj′代表粒子i在归一化后的目标空间拥挤距离,CD i,obj′代表粒子i在归一化后的决策空间拥挤距离,M表示目标空间维度,N表示决策空间维度;(3)计算粒子i的特殊拥挤距离:其中,CD avg,obj′和CD avg,dsc′分别代表归一化后的目标空间平均拥挤距离和决策空间的平均拥挤距离。
- 根据权利要求1所述的一种多目标多模态离子群优化方法,其特征在于,S4的具体步骤为:(1)创建每个粒子的档案用于存储每个粒子的历史信息,将各种群的非支配解集中的排第一位的粒子视为种群全局最佳subGbest,之后种群中的每个粒子更新自己的速度和位置,更新方法为:V′ i=wV i+c 1r 1(Pbest i-X i)+c 2r 2(Gbest-X i)X′ i=X i+V i其中,X i是当前粒子的位置,X i’是更新后粒子的位置,V i是当前粒子的速度,V i’是更新后粒子的速度,Pbest为每个粒子的个体最佳,Gbest为种群全局最佳,则为subGbest;(2)将更新后粒子存储到粒子档案中,并删除档案中被更新后粒子支配的粒子;(3)当粒子档案中有能够支配个体最佳Pbest的粒子时,则用能支配个体最佳Pbest的粒子取代个体最佳Pbest成为新的个体最佳,否则不更新个体最佳Pbest;(4)同理,当个体最佳Pbest中有能够支配全局最佳的粒子subGbest时,则用能够支配全局最佳的粒子subGbest的粒子取代全局最佳subGbest成为新的全局最佳,否则不更新全局最佳subGbest。
- 根据权利要求1所述的一种多目标多模态离子群优化方法,其特征在于,S5的具体步骤为:(1)将每个种群的非支配解集首尾相连组成一个环形拓扑,把每个非支配解集及与之相邻的两个非支配解集视为一个邻域;(2)将邻域中的所有非支配粒子放入邻域档案中,根据非支配排序法和特殊拥挤距离对粒子进行排序;(3)将邻域档案中排第一位的粒子视为邻域最佳Nbest,再更新每个种群的全局最佳subGbest,更新方法为:V′ i=wV i+c 1r 1(Pbest i-X i)+c 2r 2(Nbest i-X i)X′ i=X i+V i(4)将每个种群更新后的全局最佳subGbest放入各自的非支配解集中,将非支配解集中的粒子重新根据非支配排序法和特殊拥挤距离进行排序,删除被支配的粒子,将排第一位的粒子作为新的全局最佳subGbest。
- 根据权利要求1所述的一种多目标多模态离子群优化方法,其特征在于,S6的具体步骤为:根据S5得到的新的种群全局最佳subGbest按照S4中的规则更新种群中的粒子速度和位置,然后再按照S5中的规则更新种群全局最佳,不断循环迭代,直到满足终止条件。
- 根据权利要求1所述的一种多目标多模态离子群优化方法,其特征在于,S6的具体步骤为:迭代完成后,每个非支配解集中的非支配粒子则为决策空间中的帕累托最优解,将各帕累托最优解代入目标函数则得到在目标空间的帕累托前沿。
- 根据权利要求1所述的一种多目标多模态离子群优化方法,其特征在于,利用贝叶斯自适应共振理论将粒子群划分为若干个种群之前还包括以下步 骤:确定粒子群的相关参数,在决策空间内随机生成N个粒子,并对粒子群初始化。
- 根据权利要求9所述的一种多目标多模态离子群优化方法,其特征在于,相关参数包括粒子总数量N,粒子维度D,惯性权重w,速度控制参数c1和速度控制参数c2;初始化各个粒子的速度和位置。
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