From 1056eda83003bbe1ec5e80875157d20456be5a2a Mon Sep 17 00:00:00 2001 From: liworuo Date: Tue, 31 Dec 2024 15:38:07 +0800 Subject: [PATCH] =?UTF-8?q?fix:=204.=E6=99=BA=E8=83=BD=E6=90=9C=E7=B4=A2?= =?UTF-8?q?=E6=8A=80=E6=9C=AF=EF=BC=8C=E5=AE=8C=E5=96=84=E8=A1=A5=E5=85=85?= =?UTF-8?q?+=E6=A0=BC=E5=BC=8F=E4=BF=AE=E6=94=B9?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...34\347\264\242\346\212\200\346\234\257.md" | 233 ++++++++++++------ ...6\346\240\221\345\233\276\347\244\272.png" | Bin 0 -> 31125 bytes ...5\346\265\201\347\250\213\345\233\276.png" | Bin 0 -> 40757 bytes 3 files changed, 155 insertions(+), 78 deletions(-) create mode 100644 "\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/image/\344\270\216\346\210\226\346\240\221\345\233\276\347\244\272.png" create mode 100644 "\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/image/\350\232\201\347\276\244\347\256\227\346\263\225\346\265\201\347\250\213\345\233\276.png" diff --git "a/\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/4. \346\231\272\350\203\275\346\220\234\347\264\242\346\212\200\346\234\257.md" "b/\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/4. \346\231\272\350\203\275\346\220\234\347\264\242\346\212\200\346\234\257.md" index 5989953..4d37a61 100644 --- "a/\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/4. \346\231\272\350\203\275\346\220\234\347\264\242\346\212\200\346\234\257.md" +++ "b/\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/4. \346\231\272\350\203\275\346\220\234\347\264\242\346\212\200\346\234\257.md" @@ -2,61 +2,119 @@ ## 搜索的含义 -- 搜索:依靠经验,利用已有知识,根据问题的实际情况,不断寻找可利用知识,从而构造一条代价最小的推理路线,使问题得以解决的过程称为搜索 -- 智能搜索:是指可以利用搜索过程得到的中间信息来引导搜索项往最优方向发展的算法 +### 搜索 + +依靠经验,利用已有知识,根据问题的实际情况,不断寻找可利用知识,从而构造一条代价最小的推理路线,使问题得以解决的过程称为搜索 + +### 智能搜索 + +是指可以利用搜索过程得到的中间信息来引导搜索项往最优方向发展的算法 + +
+ +智能搜索类型 +- 基于搜索空间的类型 + - 状态空间 (A 算法、A-star 算法) + - 与或树 (问题规约法) + - 博弈树 (极大/极小算法、$\alpha-\beta 剪枝$) +- 基于随机算法的类型 + - 演化机制 (遗传算法) + - 种群寻优 (蚁群算法、粒子群算法 +) + +
## 状态空间问题求解方法 ### 状态空间问题表示 -- 状态:是表示问题求解过程中每一步问题状况的数据结构,它可形式地表示为: $S_1 = \set{ S_{k0}, S_{k1}, \cdots}$ 。当对每一个分量都给以确定的值时,就得到了一个具体的状态 -- 操作:也成为算符,它是把问题从一种状态变为另一种状态的手段。操作可以是一个机械步骤,一个计算,一条规则或一个过程。操作可理解为状态集合上的一个函数,它描述了状态之间的关系 +#### 状态 (State) + +是表示问题求解过程中每一步问题状况的数据结构,它可形式地表示为 + +$$ +S_{1}={S_{k0},S_{k1},\cdots} +$$ + +当对每一个分量都给以确定的值时,就得到了一个具体的状态 + +#### 操作 (Operator) + +也成为算符,它是把问题从一种状态变为另一种状态的手段 + +操作可以是一个机械步骤,一个计算,一条规则或一个过程。操作可理解为状态集合上的一个函数,它描述了状态之间的关系 -- 状态空间:用来描述一个问题的全部状态以及这些状态之间的相互关系。常用一个三元组表示为:(S,F,G)。其中 S 为问题的所有初始状态集合;F 为操作的集合;G 为目标状态的集合。状态空间也可以用一个赋值的有向图来表示,该有向图称为状态空间图。 +#### 状态空间 (State space) + +用来描述一个问题的全部状态以及这些状态之间的相互关系,常用一个三元组表示为 + +$$ +(S,F,G) +$$ + +其中 $S$ 为问题的所有初始状态集合;$F$ 为操作的集合;$G$ 为目标状态的集合 + +状态空间也可以用一个赋值的有向图来表示,该有向图称为状态空间图 ### 状态空间问题求解 -- 首先,为问题选择适当的”状态“及”操作“的形式化描述方法 -- 然后,从某个初始状态出发,每次使用一个”操作“,递增地建立起操作序列,直到达到目标状态为止 -- 最后,由初始状态到目标状态所使用地算符序列就是该问题的一个解 +- 首先,为问题选择适当的 "状态" 及 "操作" 的形式化描述方法 +- 然后,从某个初始状态出发,每次使用一个 "操作",递增地建立起操作序列,直到达到目标状态为止 +- 最后,由初始状态到目标状态所使用地**算符序列**就是该问题的一个解 ## 问题规划求解方法 -### 问题的分解与等价变换 +### 基本思想 + +当一问题较复杂时,可通过**分解**或**变换**,将其转化为一系列较简单的子问题,然后通过对这些子问题的求解来实现对原问题的求解 + +### 分解 ("与") + +一个问题 $P$ 可以规约为一组子问题 $P_1,P_2,…P_n$ -- 基本思想:当一问题较复杂时,可通过分解或变换,将其转化为一系列较简单的子问题,然后通过对这些子问题的求解来实现对原问题的求解 +并且只有当所有子问题 $P_i$ 都有解时原问题才有解,任何一个子问题无解都会导致原问题$P$无解,则称此种规约为问题的分解 -- 分解:如果一个问题 $P$ 可以规约为一组子问题 $P_1,P_2,…P_n$ ,并且只有当所有子问题 $P_i$ 都有解时原问题才有姐,任何一个子问题无解都会导致原问题 $P$ 无解,则称此种规约为问题的分解。即分解所得到的子问题的"与"与原问题 $P$ 等价 +**分解**所得到的子问题的“与”与原问题 $P$ 等价 -- 等价变换:如果一个问题 $P$ 可以归约为一组子问题 $P_1,P_2,\cdots ,P_n$ ,并且子问题 $P_i$ 中只要有一个有解则原问题就有解,只有当所有子问题都无解时原问题才无解,称此种规约为问题的等价变换,简称变换。即变换所得到的子问题的”或”与原问题等价 +### 等价变换 ("或") + +一个问题 $P$ 可以归约为一组子问题 $P_1,P_2,\cdots ,P_n$ + +并且子问题 $P_i$ 中只要有一个有解则原问题就有解,只有当所有子问题都无解时原问题才无解,称此种规约为问题的等价变换,简称变换 + +**变换**所得到的子问题的 "或" 与原问题等价 ### 问题规约的与/或树表示 - ![image-20241226102939172](./image/与或树图示.png) + ![与或树图示](./image/与或树图示.png) + +#### 本原问题 -- 本原问题是指那种不能 (或不需要) 再进行分解或变换且可以直接解答的问题 +是指那种不能 (或不需要) 再进行分解或变换且可以直接解答的问题 #### 端节点与终止节点 -- 端节点是在与/或树种没有子节点的节点;本原问题所对应的节点称为终止节点。 +端节点是在与/或树种没有子节点的节点;本原问题所对应的节点称为终止节点。 > [!NOTE] > -> 终止节点一定是端节点,但段节点却不一定是终止节点。原因是,本原节点要做到的条件很难,所以其对应的终止节点更难达成 +> 终止节点一定是端节点,但段节点却不一定是终止节点 +> +> 原因是,本原节点要做到的条件很难,所以其对应的终止节点更难达成 +> +> 终止节点都是可解节点 #### 可解节点与不可解节点 **定义可解节点**: -> [!NOTE] -> -> 满足以下三个条件之一的节点为可解节点 +满足以下三个条件之一的节点为可解节点 - 任何终止节点都是可解节点 - 对“或”节点,当其子节点至少有一个为可解节点时,则该或节点就是可解节点 - 对“与”节点,只有当其子节点全部为可解节点时,该与节点才是可解节点 -**定义不可解节点:** +**定义不可解节点:**(与上类似) - 不为终止节点的端节点是不可解节点 - 对“或”节点,若其全部子节点都是不可解节点,则该或节点是不可解节点 @@ -64,7 +122,9 @@ #### 解树 -由可解节点构成,并且由这些可解节点可以推出初始节点为可解节点的子树为解树。在解树中一定包含初始节点。问题规约求解过程就实际上就是生成解树,即证明原始节点是可解节点的过程 +由可解节点构成,并且由这些可解节点可以推出初始节点为可解节点的子树为解树。在解树中一定包含初始节点 + +问题规约求解过程就实际上就是生成解树,即证明原始节点是可解节点的过程 ## 进化搜索法概述 @@ -94,10 +154,8 @@ 遗传将自身基因信息遗传给下一代,由于遗传的作用,自然界才能有稳定的物种 -- 细胞(构成基本结构与功能的单位) - +- 细胞 (构成基本结构与功能的单位) - 染色体 (包含所有遗传信息的复杂而又微小的丝状化合物) - - 基因 (组成遗传信息,决定生物性状,使遗传的基本单位) - DNA (脱氧核糖核酸,构成染色体的主要物质) - 基因组 (一个细胞中的所有染色体所携带的遗传信息的全体称为一个基因组) @@ -124,7 +182,7 @@ 假设 $P$ 为种群, $t$ 为进化代数, $P(t)$ 为第 $t$ 代种群,则进化计算的基本结构可粗略描述如下: ```markdown -{ +{ 确定编码形式并生成搜索空间; 初始化各个进化参数,并设置进化代数 t=0; 初始化种群 P(0); @@ -166,6 +224,10 @@ $$ > [!NOTE] > > 八数码难题 +> +> 计算值 = 深度 + 异位数 +> +> S3 节点,深度为 3,异位数为 1(跟 $S_g$ 比较不一样节点的个数);S1 节点红线处,深度为 1,异位数位 3 (PPT 上的例题) ## A 算法 @@ -189,21 +251,6 @@ $$ - 根据各节点的估价函数值,对 Open 表中的全部节点按从小到大的顺序重新进行排序 - 转第 (2) 步 -#### 八数码难题 - -```bash -S0 Sg -2 8 3 1 2 3 -1 4 8 4 -7 6 5 9 6 5 -``` - -![image-20241226140912477](./image/八数码难题.png) - -计算值 = 深度 + 异位数 - -S3 节点,深度为 3,异位数为 1(跟 $S_g$ 比较不一样节点的个数);S1 节点红线处,深度为 1,异位数位 3 - # 与/或树的启发式搜索 与/或树的启发式搜索过程实际上是一种利用搜索过程所得到的启发性信息寻找最优解树的过程 @@ -238,58 +285,71 @@ $$ ### 博弈 -概念:是一类具有智能行为的竞争活动,如下棋、战争等 +是一类具有智能行为的竞争活动,如下棋、战争等 -类型: +#### 类型 -- 双人完备信息博弈:两位选手 (MAX、MIN) 对垒,轮流走步,每一方不仅知道对方已经走过的棋步,而且还能估计出对方未来的走步 -- 机遇性博弈:存在不可预测性的博弈 +**双人完备信息博弈**:两位选手 (MAX、MIN) 对垒,轮流走步,每一方不仅知道对方已经走过的棋步,而且还能估计出对方未来的走步 -博弈树:把双人完备信息博弈过程用图表示出来,就得到一颗与/或树,这种与/或树被称为博弈树。在博弈树中,那些下一步该 MAX 走步的节点称为 MAX 节点,下一步该 MIN 走步的节点称为 MIN 节点 +**机遇性博弈**:存在不可预测性的博弈 -### 博弈的特点 +#### 博弈树 + +把双人完备信息博弈过程用图表示出来,就得到一颗与/或树,这种与/或树被称为博弈树。在博弈树中,那些下一步该 MAX 走步的节点称为 MAX 节点,下一步该 MIN 走步的节点称为 MIN 节点 -博弈树的特点: +### 博弈的特点 - 博弈的初始状态是初始节点 -- 博弈树中的或节点和与节点是逐层交替出现的 +- 博弈树中的 "或" 节点 "和" 与节点是逐层交替出现的 - 整个博弈过程始终站在某一方的立场上,例如 MAX 方。所有能使自己一方获胜的终局都是本原问题,相应的节点是可解节点;所有使对方获胜的终局都是不可解节点 ## 极大极小过程 对于复杂的博弈问题,可以用当前正在考察的节点生成一棵部分博弈树,并用估价函数 $f(n)$ 对叶节点进行静态估值 -- 对叶节点的估值方法是:对 MAX 有利的节点,其估价函数取正值;那些对 MIN 有利的节点,其估价函数取负值;那些使双方均等的节点,其估价函数取接近于 0 的值 -- 为非叶节点的值,必须从叶节点开始向上倒推 - - 对于 MAX 节点,由于 MAX 方总是选择估值最大的走步,因此 MAX 节点的倒退值应该取其后继节点估值的最大值 - - 对于 MIN 节点,由于 MIN 方总是选择估值最小的走步,因此 MIN 节点的倒退值应该取其后继节点估值的最小值 - - 这样一步一步的计算倒推值,直至求出初始节点的倒推值为止。这一过程称为极大极小过程 +### 对叶节点的估值方法 -# 进化搜索 +对 MAX 有利的节点,其估价函数取正值;那些对 MIN 有利的节点,其估价函数取负值;那些使双方均等的节点,其估价函数取接近于 0 的值 -## 蚁群算法 +### 非叶节点的值 + +必须从叶节点开始向上倒推 +- 对于 MAX 节点,由于 MAX 方总是选择估值最大的走步,因此 MAX 节点的倒退值应该取其后继节点估值的最大值 +- 对于 MIN 节点,由于 MIN 方总是选择估值最小的走步,因此 MIN 节点的倒退值应该取其后继节点估值的最小值 +- 这样一步一步的计算倒推值,直至求出初始节点的倒推值为止。这一过程称为极大极小过程 + +# 进化搜索 ### 群智能 -人们把群居昆虫的集体行为称作“群智能” +人们把群居昆虫的集体行为称作“群智能” (“群体智能”、 “群集智能”、“集群智能”等) + +#### 特点 + +个体的行为很简单,但当他们一起协同工作时,**却能够突现出非常复杂的行为特征** + +#### 描述 -特点:个体的行为很简单,但当他们一起协同工作时,却能够突现出非常复杂的行为特征 +群智能作为一种新兴的演化计算技术已称为研究焦点,它与人工生命,特别是进化策略以及遗传算法有着极为特殊的关系 -描述:群智能作为一种新兴的演化计算技术已称为研究焦点,它与人工生命,特别是进化策略以及遗传算法有着极为特殊的关系 +#### 特性 -特性:无智能的主题通过合作表现出智能行为的特性,在没有集中控制且不提供全局模型的前提下,为寻找复杂的分布式问题求解方案提供了基础。 +无智能的主题通过合作表现出智能行为的特性,在没有集中控制且不提供全局模型的前提下,为寻找复杂的分布式问题求解方案提供了基础 -优点: +#### 优点 - 灵活性:群体可以适应随时改变的环境 -- 稳健性:即使个体失败,整个群体仍能完成任务;自我组织:活动既不受中央控制,也不受局部监管 +- 稳健性:即使个体失败,整个群体仍能完成任务 +- 自我组织:活动既不受中央控制,也不受局部监管 -典型算法 +#### 典型算法 - 蚁群算法 - 粒子群优化算法 - 人工蜂群算法 +## 蚁群算法 + ### 蚁群优化算法原理 - 蚁群的自组织行为 @@ -317,20 +377,19 @@ $$ ### 基本蚁群优化算法 -> [!NOTE] -> -> 解决 TSP 问题 +#### 解决 TSP 问题思想 假设蚁群算法中的每只蚂蚁都是具有下列特征的简单智能体 -1. 每次周游,每只蚂蚁在其经过地支路 (i,j) 上都留下信息素 - +1. 每次周游,每只蚂蚁在其经过地支路 $(i,j)$ 上都留下信息素 2. 蚂蚁选择城市的概率与城市之间的距离和当前连接支路上所包含的信息素余量有关 3. 为了加强蚂蚁进行合法的周游,直到一次周游完成后,才允许蚂蚁游走已访问的城市,可以用禁忌表来控制 +#### 解决 TSP 问题算法步骤 + - 在算法的初始时刻,将 $m$ 只蚂蚁随机放到 $n$ 座城市 -- 将每只蚂蚁 $k$ 的禁忌表 $tabu_k(s)$ 的第一个元素 $tabu_k(1)$ 设置为它当前所在城市 -- s 设各路径上的信息素 $\tau_{ij}(0)=C$ ( $C$ 为一较小的常数) +- 将每只蚂蚁 $k$ 的禁忌表 $tabu_k(s)$ 的第一个元素 $tabu_k(1)$ 设置为它当前所在城市; +- s 设各路径上的信息素 $\tau_{ij}(0)=C$ ( $C$ 为一较小的常数) ; - 每只蚂蚁根据路径上的信息素和启发式信息 (两城市间的距离) 独立地选择下一座城市,在时刻 $t$ ,蚂蚁 $k$ 从城市 $i$ 转移到城市 $j$ 的概率为 $$ @@ -341,11 +400,13 @@ p_{ij}^k(t)= \end{cases} $$ +$\alpha$ 、 $\beta$ 分别表示信息素和启发式因子的相对重要程度 + $$ J_k(i)=\set{1,2,\cdots,n}-tabu_k $$ -当所有蚂蚁完成一次周游后,各路径上的信息素将进行更新 +- 当所有蚂蚁完成一次周游后,各路径上的信息素将进行更新 $$ \tau_{ij}(t+n)=(1-\rho)\tau_{ij}(t)+\Delta\tau_{ij} @@ -362,7 +423,15 @@ $$ 其中 $\rho$ 代表路径上信息素的蒸发系数;Q 为正常数,表示蚂蚁循环一周或一个过程在经过的路径上所释放的信息素总量,影响算法的收敛速度; $L_k$ 表示第 $k$ 只蚂蚁在本次周游中所走过的路径长度 -#### 三种模型 +
+ +蚁群算法流程图 + +![蚁群算法流程图](./image/蚁群算法流程图.png) + +
+ +#### 三种模型 (信息素) - 蚁周,全局信息,所有蚂蚁都完成了自己的行程后才对信息素进行更新 - 蚁量,局部信息,两个位置节点间每移动一次就更新信息素 @@ -374,7 +443,9 @@ $$ \frac{Q}{L_{k}}, & \text{蚂蚁 } k \text{ 在本次周游中经过 } ij \\ 0, & \text{否则} \end{cases} -$$ +$$ + +$L_k$ 表示第 $k$ 只蚂蚁在本次周游中所走过的路径长度 $$ \Delta \tau_{ij}^k = @@ -382,7 +453,9 @@ $$ \frac{Q_{2}}{d}, & \text{蚂蚁 } k \text{ 在时刻 } t + 1 \text{ 经过 } ij \\ 0, & \text{否则} \end{cases} -$$ +$$ + +$d$ 表示第 $k$ 只蚂蚁在两个位置节点间所走过的路径长度 $$ \Delta \tau_{ij}^k = @@ -392,32 +465,36 @@ Q, & \text{蚂蚁 } k \text{ 在时刻 } t \text{ 经过 } ij \\ \end{cases} $$ +### 蚁群优化算法的参数设置和基本属性 + #### 参数 $\alpha,\beta$ 对算法性能的影响 -停滞现象:所有蚂蚁都选择相同的路径,即系统不再搜索更好的解。原因在于较好路径上的信息素远大于其他边上的,从而使所有蚂蚁都选择该路径 +停滞现象 ( $\alpha$ ):所有蚂蚁都选择相同的路径,即系统不再搜索更好的解。原因在于较好路径上的信息素远大于其他边上的,从而使所有蚂蚁都选择该路径 - $\alpha$ 取较大值时,意味着在选择路径时,路径上的信息素非常重要 - $\alpha$ 取较小值时,变成随机的贪婪算法 - $\alpha$ 取 0 时,蚂蚁之间无协同作用 - $\alpha$ 取 1 时,有协同作用 -##### 蚂蚁数 $m$ 对算法的影响 +#### 蚂蚁数 $m$ 对算法的影响 - 当 $n=1m-1.5m$ 时,蚂周可以有较好的全局收敛性和收敛速度 -##### 蚂蚁的初始分布 +#### 蚂蚁的初始分布 蚂蚁分布在不同城市中,可以获得较高性能 +在不同城市分布时,随机分布与统一分布的差别不大 + ### 蚁群算法优点与不足 -##### 优点 +#### 优点 - 较强的鲁棒性 - 分布式计算 - 易于与其他方式结合 -##### 缺点 +#### 缺点 - 计算量大、搜索时间较长 - 容易出现停止现象 diff --git "a/\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/image/\344\270\216\346\210\226\346\240\221\345\233\276\347\244\272.png" "b/\344\272\272\345\267\245\346\231\272\350\203\275\345\237\272\347\241\200\344\270\216\345\272\224\347\224\250/image/\344\270\216\346\210\226\346\240\221\345\233\276\347\244\272.png" new file mode 100644 index 0000000000000000000000000000000000000000..3c2991f5052a884621ac4f9762dc8b18dd0f6956 GIT binary patch literal 31125 zcmX_IbyOAI+oijaZWQV6PDx4W?v%WANOwsHNQVd#m+tQF=8}SRcl#TBzxDmWV$Cq; zIcJ{OCuVOVRg`4VQHW8XprFv@WTn)gpkN}PprGTC5P&=DKd5qmU(g^m83`!xIO#rc z0cRzyC=LZx6_5I40uNjxJIU&Rpr9~%UVoqm9E;7Npu8C5q{P)d4Ubn5UEf%cz;C2t zfj3ZjmR3ecXSP*qh??*${lM49HvYW%oW3c6Yo+{}5j zeE%_QbxGJKsJp zRm+{MG+na`(SmkHktgcr&xyygg)=HyibG(OiV}IRhzavel-PyV()pN@RmhMtTEef#DL3 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