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第42行： − A basic variant of the PSO algorithm works by having a population (called a swarm) of [[candidate solution]]s (called particles). These particles are moved around in the search-space according to a few simple formulae.<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications|journal=Mathematical Problems in Engineering|date=2015|volume=2015|page=931256|url=http://www.hindawi.com/journals/mpe/2015/931256}}</ref> The movements of the particles are guided by their own best known position in the search-space as well as the entire swarm's best known position. When improved positions are being discovered these will then come to guide the movements of the swarm. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered. 第50行：
第49行： − − Formally, let ''f'': ℝ<sup>''n''</sup> → ℝ be the cost function which must be minimized. The function takes a candidate solution as an argument in the form of a [[Row vector|vector]] of [[real number]]s and produces a real number as output which indicates the objective function value of the given candidate solution. The [[gradient]] of ''f'' is not known. The goal is to find a solution '''a''' for which ''f''('''a''') ≤ ''f''('''b''') for all '''b''' in the search-space, which would mean '''a''' is the global minimum. 第57行：
第54行： − − − Let ''S'' be the number of particles in the swarm, each having a position '''x'''<sub>i</sub> ∈ ℝ<sup>''n''</sup> in the search-space and a velocity '''v'''<sub>i</sub> ∈ ℝ<sup>''n''</sup>. Let '''p'''<sub>i</sub> be the best known position of particle ''i'' and let '''g''' be the best known position of the entire swarm. A basic PSO algorithm is then:<ref name=clerc12spso/> 第66行：
第60行： − − <!-- Please see discussion page why this particular PSO variant was chosen. --> 第168行：
第160行： − − The values '''b<sub>lo</sub>''' and '''b<sub>up</sub>''' represents the lower and upper boundaries of the search-space. The termination criterion can be the number of iterations performed, or a solution where the adequate objective function value is found.<ref name=bratton2007/> The parameters ω, φ<sub>p</sub>, and φ<sub>g</sub> are selected by the practitioner and control the behaviour and efficacy of the PSO method, see [[#Parameter selection|below]].
→Algorithm
== Algorithm ==
== Algorithm ==
算法<br>
算法<br>
A basic variant of the PSO algorithm works by having a population (called a swarm) of candidate solutions (called particles). These particles are moved around in the search-space according to a few simple formulae. The movements of the particles are guided by their own best known position in the search-space as well as the entire swarm's best known position. When improved positions are being discovered these will then come to guide the movements of the swarm. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered.
A basic variant of the PSO algorithm works by having a population (called a swarm) of candidate solutions (called particles). These particles are moved around in the search-space according to a few simple formulae. The movements of the particles are guided by their own best known position in the search-space as well as the entire swarm's best known position. When improved positions are being discovered these will then come to guide the movements of the swarm. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered.
==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]）“粒子群优化算法的一个基本变种是通过一个候选解(称为粒子)的群体来工作”，这句话里面的population和swarm有点区分不开。
==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]）“粒子群优化算法的一个基本变种是通过一个候选解(称为粒子)的群体来工作”，这句话里面的population和swarm有点区分不开。
Formally, let f: ℝ<sup>n</sup> → ℝ be the cost function which must be minimized. The function takes a candidate solution as an argument in the form of a vector of real numbers and produces a real number as output which indicates the objective function value of the given candidate solution. The gradient of f is not known. The goal is to find a solution a for which f(a) ≤ f(b) for all b in the search-space, which would mean a is the global minimum.
Formally, let f: ℝ<sup>n</sup> → ℝ be the cost function which must be minimized. The function takes a candidate solution as an argument in the form of a vector of real numbers and produces a real number as output which indicates the objective function value of the given candidate solution. The gradient of f is not known. The goal is to find a solution a for which f(a) ≤ f(b) for all b in the search-space, which would mean a is the global minimum.
形式上，设 ''f'': ℝ<sup>''n''</sup> → ℝ是需要最小化的'''<font color="#ff8000">成本函数 Cost Function</font><font>'''。该函数将一个候选解作为实数向量形式的参数，并产生一个实数作为输出，该实数表示给定候选解时目标函数的值。 ''f''的梯度并不知道。目标是找到搜索空间中所有 '''b'''对应的''f''('''a''') ≤ ''f''('''b''')的解'''a'''，这意味着 a 是全局最小值。
形式上，设 ''f'': ℝ<sup>''n''</sup> → ℝ是需要最小化的'''<font color="#ff8000">成本函数 Cost Function</font><font>'''。该函数将一个候选解作为实数向量形式的参数，并产生一个实数作为输出，该实数表示给定候选解时目标函数的值。 ''f''的梯度并不知道。目标是找到搜索空间中所有 '''b'''对应的''f''('''a''') ≤ ''f''('''b''')的解'''a'''，这意味着 a 是全局最小值。
Let S be the number of particles in the swarm, each having a position x<sub>i</sub> ∈ ℝ<sup>n</sup> in the search-space and a velocity v<sub>i</sub> ∈ ℝ<sup>n</sup>. Let p<sub>i</sub> be the best known position of particle i and let g be the best known position of the entire swarm. A basic PSO algorithm is then:
Let S be the number of particles in the swarm, each having a position x<sub>i</sub> ∈ ℝ<sup>n</sup> in the search-space and a velocity v<sub>i</sub> ∈ ℝ<sup>n</sup>. Let p<sub>i</sub> be the best known position of particle i and let g be the best known position of the entire swarm. A basic PSO algorithm is then:
<!-- Please see discussion page why this particular PSO variant was chosen. -->
<!-- Please see discussion page why this particular PSO variant was chosen. -->
更新粒子最优位置为: '''g''' ← '''p'''<sub>i</sub>
更新粒子最优位置为: '''g''' ← '''p'''<sub>i</sub>
The values b<sub>lo</sub> and b<sub>up</sub> represents the lower and upper boundaries of the search-space. The termination criterion can be the number of iterations performed, or a solution where the adequate objective function value is found. The parameters ω, φ<sub>p</sub>, and φ<sub>g</sub> are selected by the practitioner and control the behaviour and efficacy of the PSO method, see below.
The values b<sub>lo</sub> and b<sub>up</sub> represents the lower and upper boundaries of the search-space. The termination criterion can be the number of iterations performed, or a solution where the adequate objective function value is found. The parameters ω, φ<sub>p</sub>, and φ<sub>g</sub> are selected by the practitioner and control the behaviour and efficacy of the PSO method, see below.