【最优化导论】粒子群优化
粒子群优化并不是只更新单个迭代点 https://www.zhihu.com/equation?tex=x%5E%7B%28k%29%7D ，而是更新一组迭代点，称为群。群中每个点是一个粒子，可以将群视为一个无序的群体，其中每个成员都在移动，意在形成聚集，但移动方向是随机的。
每个粒子都持续追踪当前为止的最好位置。对某个粒子相关的当前为止的最好位置为 https://www.zhihu.com/equation?tex=pbest ，全局当前位置最好的位置为 https://www.zhihu.com/equation?tex=gbest 。
引入Hadamard积算子， https://www.zhihu.com/equation?tex=A%5Ccirc+B 表示A和B各元素对应相乘得到的矩阵。
算法：
[*] 令https://www.zhihu.com/equation?tex=k%3A%3D0 ，随机产生一个初始的粒子群，即产生d个粒子的位置 https://www.zhihu.com/equation?tex=x_i%5E%7B%280%29%7D 及其对应的速度 https://www.zhihu.com/equation?tex=v_i%5E%7B%280%29%7D ， https://www.zhihu.com/equation?tex=p_i%5E%7B%280%29%7D%3Dx_i%5E%7B%280%29%7D%2Ci%3D1%2C2%2C%5Cdots%2Cd ，令 https://www.zhihu.com/equation?tex=g%5E%7B%280%29%7D%3Darg%5Cmin_%7Bx%5Cin+%5C%7Bx_1%5E%7B%280%29%7D%2C%5Cdots%2Cx_d%5E%7B%280%29%7D%5C%7D+%7Df%28x%29 。
[*]针对每个 https://www.zhihu.com/equation?tex=i%3D1%2C2%2C%5Cdots%2Cd ，随机生成两个n为向量 https://www.zhihu.com/equation?tex=r_i%5E%7B%28k%29%7D%2Cs%5E%7B%28k%29%7D_i ，按照均匀分布的原则抽取区间(0,1)中的随机数，构成这两个向量的元素，令： https://www.zhihu.com/equation?tex=v_i%5E%7B%28k%2B1%29%7D%3D%5Comega+v_i%5E%7B%28k%29%7D%2Bc_1r_i%5E%7B%28k%29%7D%5Ccirc%28p_i%5E%7B%28k%29%7D-x_i%5E%7B%28k%29%7D%29%2Bc_2s_i%5E%7B%28k%29%7D%5Ccirc+%28g%5E%7B%28k%29%7D-x_i%5E%7B%28k%29%7D%29%5C%5Cx_i%5E%7B%28k%2B1%29%7D%3Dx_i%5E%7B%28k%29%7D%2Bv_i%5E%7B%28k%2B1%29%7D+
[*]针对每个 https://www.zhihu.com/equation?tex=i%3D1%2C%5Cdots%2Cd ，如果 https://www.zhihu.com/equation?tex=f%28x_i%5E%7B%28k%2B1%29%7D%29%3Cf%28p_i%5E%7B%28k%29%7D%29 ，令 https://www.zhihu.com/equation?tex=p_i%5E%7B%28k%2B1%29%7D%3Dx_i%5E%7B%28k%2B1%29%7D ，否则令 https://www.zhihu.com/equation?tex=p_i%5E%7B%28k%2B1%29%7D%3Dp_i%5E%7B%28k%29%7D
[*]如果存在 https://www.zhihu.com/equation?tex=i%5Cin%5C%7Bi%2C%5Cdots%2Cd%5C%7D ，使 https://www.zhihu.com/equation?tex=f%28x_i%5E%7B%28k%2B1%29%7D%29%3Cf%28g%5E%7B%28k%29%7D%29 ，令 https://www.zhihu.com/equation?tex=i%5E%2A%3Darg%5Cmin_i+f%28x%5E%7B%28k%2B1%29%7D_i%29 ， https://www.zhihu.com/equation?tex=g%5E%7B%28k%2B1%29%7D%3Dx_%7Bi%2A%7D%5E%7B%28k%2B1%29%7D ，否则令 https://www.zhihu.com/equation?tex=g%5E%7B%28k%2B1%29%7D%3Dg%5E%7B%28k%29%7D
[*]如果满足停止条件，就停止迭代
[*]令 https://www.zhihu.com/equation?tex=k%3A%3Dk%2B1 ，回到第2步
参数w表示惯性参数，取稍微小于1的值， https://www.zhihu.com/equation?tex=c_1%2Cc_2 决定了粒子趋向于”好位置“的程度，分别表示来自”认知“和”社会“部分的影响因素，即粒子本身最好位置和全局最好位置对其运动的影响，建议取值 https://www.zhihu.com/equation?tex=c_1%3Dc_2%5Capprox+2
粒子群优化算法的变种
在收敛因子粒子群优化算法，速度更新公式为 https://www.zhihu.com/equation?tex=v_i%5E%7B%28k%2B1%29%7D%3D%5Cmathcal%7BK%7D%28v_i%5E%7B%28k%29%7D%2Bc_1r_i%5E%7B%28k%29%7D%5Ccirc%28p_i%5E%7B%28k%29%7D-x_i%5E%7B%28k%29%7D%29%2Bc_2s_i%5E%7B%28k%29%7D%5Ccirc%28g%5E%7B%28k%29%7D-x_i%5E%7B%28k%29%7D%29%29
https://www.zhihu.com/equation?tex=%5Cmathcal%7BK%7D 为收敛系数： https://www.zhihu.com/equation?tex=%5Cmathcal%7BK%7D%3D%5Cfrac%7B2%7D%7B2-%5Cphi-%5Csqrt%7B%5Cphi%5E2-4%5Cphi%7D%7D
收敛系数的作用在于加快收敛。 https://www.zhihu.com/equation?tex=%5Cphi%3Dc_1%2Bc_2%2C%5Cphi%3E4 。
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