From 685ab593db0bd87e9246d7ab1ebaf5be8d000836 Mon Sep 17 00:00:00 2001 From: zhaoxi <535394140@qq.com> Date: Sat, 4 May 2024 02:42:57 +0800 Subject: [PATCH] =?UTF-8?q?=E5=AE=8C=E6=88=90=E4=B8=80=E7=BB=B4=E5=92=8C?= =?UTF-8?q?=E5=A4=9A=E7=BB=B4=E6=97=A0=E7=BA=A6=E6=9D=9F=E9=9D=9E=E7=BA=BF?= =?UTF-8?q?=E6=80=A7=E4=BC=98=E5=8C=96=E7=9A=84=E5=8A=9F=E8=83=BD=EF=BC=8C?= =?UTF-8?q?=E5=B9=B6=E6=B7=BB=E5=8A=A0=E6=96=87=E6=A1=A3?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- doc/img/fminbnd_forward.png | Bin 0 -> 2312 bytes doc/img/fminbnd_forward_2.png | Bin 0 -> 2559 bytes doc/img/fminbnd_principle.png | Bin 0 -> 2480 bytes doc/img/ls-eg1.png | Bin 3259 -> 2310 bytes doc/img/ls-eg3.png | Bin 26705 -> 13492 bytes doc/img/ls-equations.png | Bin 27519 -> 21498 bytes doc/img/ls-line-vertical.png | Bin 8799 -> 1582 bytes doc/img/ls-vector-vertical.png | Bin 8163 -> 1739 bytes doc/img/region_reduce.png | Bin 0 -> 18775 bytes doc/img/upper_base.png | 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z5GFz@PrhK6-I(@JI2yr*7Sip&tn*i)t}|#ayHaC-1{MNL$XIy)dd`AuzMEcK@Qtpk@xzKbJ)Gffq*9&^%9D%uC1* zx5nkZZL>hU3l;JuovYQnosUX99|d0zV|Yg~T-6}eAhz=?un6m1D_w`ICl9|G796xztupM?F++Cg{O4GMGL5hcJewVDo6_;> zCD!hu0^J-1`=EK$LyNV=vjf&RO$ni+hwt<6s~)2>Tblp8c?c?OnMh$n8pvG4+9TAj zbi?qGA*e&3kbl7rg?*_u-B9Zu+EjZ6aK2#osrNaY1jb8_`P;>F?*)?MLHm{lCrYc| zHe$G{Jj87{W2GUw>bR!E3BH2twq!6io%+#8D(T4Mx~brJ@`njI#Z3hN$(*ncQf6In|_e*L1p z{TGhKYCg>VZ2O-VW|vF7iCG&%k8eE{q{H>((cQNPPj9dZo!XO@R@GRrMIatkeK+&D zUWCPnF!}SvzV-eJ$jZkt)mHh{f3ZLRCAh@-?|1+6(*N_jA&!9=p6n0SY~@WKO-wA# zw6oui00Sb}Kg^O^SkL~7Qy9wE?bP-Rs0f-!F`f$ctu>*s9}DhAEm4~NACQ_LbqP|7 zW6*>YFf<}BrfMV8w363OWp_My^{XcYdz(nfA前向差商,\f$\fml{1-2b}\f$称为后向差商。由于截断误差的存在,差商法的精度不高,因此我们可以考虑使用更高阶的差商法。我们将\f$\fml{1-1a}\f$和\f$\fml{1-1b}\f$相减,得到: + +\f[f(x+h)-f(x-h)=2f'(x)h+\frac2{3!}f'''(\xi)h^3+o(h^3)\tag{1-3a}\f] + +整理得到 + +\f[f'(x)=\frac{f(x+h)-f(x-h)}{2h},\qquad R=-\frac{f'''(\xi)}{3!}h^2+o(h^2)\tag{1-3b}\f] + +如果导数\f$f'(x)\f$用\f$\fml{1-3b}\f$计算,那么截断误差为\f$o(h^2)\f$,比\f$\fml{1-2a}\f$和\f$\fml{1-2b}\f$的截断误差\f$O(h)\f$更小。这种方法称为中心差商。 + +### 2. Richardson 外推原理 + +#### 2.1 公式推导 + +将公式\f$\fml{1-3b}\f$扩展,可以得到更完全的写法: + +\f[f'(x)=\green{\frac{f(x+h)-f(x-h)}{2h}}-\left[\frac1{3!}f^{(3)}(x)h^2+\frac1{5!}f^{(5)}(x)h^4+\frac1{7!}f^{(7)}(x)h^6+\cdots\right]\tag{2-1}\f] + +一般的,可以记作 + +\f[L=\green{\varphi(h)}+\left[a_2h^2+a_4h^4+a_6h^6+\cdots\right]\tag{2-2}\f] + +正如上一节所说,如果\f$L\f$用\f$\varphi(h)\f$来近似表示,则截断误差是按\f$h^2\f$展开的幂级数,误差阶为\f$o\left(h^2\right)\f$。 + +事实上,对于公式\f$\fml{2-2}\f$,求解\f$L\f$的过程还可以继续向前推进(外推),不妨以\f$\frac h2\f$替换\f$\fml{2-2}\f$中的\f$h\f$,可得 + +\f[L=\varphi\left(\frac h2\right)+\left[a_2\frac{h^2}4+a_4\frac{h^4}{16}+a_6\frac{h^6}{64}+\cdots\right]\tag{2-3}\f] + +由\f$4\times\fml{2-3}-\fml{2-2}\f$可以得到 + +\f[L=\left[\frac43\varphi\left(\frac h2\right)-\frac13\varphi(h)\right]-\left[a_4\frac{h^4}4+5a_6\frac{h^6}{16}+\cdots\right]\tag{2-4}\f] + +上式表达了外推过程的第 1 步,说明\f$\varphi(h)\f$和\f$\varphi\left(\frac h2\right)\f$的一个线性组合提供了导数即\f$L\f$的新的计算公式,其精度提高至了\f$o\left(h^4\right)\f$ + +继续进行第 2 步,令 + +\f[\psi(h)=\frac43\varphi\left(\frac h2\right)-\frac13\varphi(h)=\frac1{4^1-1}\left[4^1\varphi\left(\frac h2\right)-\varphi(h)\right]\tag{2-5}\f] + +那么\f$\fml{2-4}\f$可以改写成 + +\f[L=\psi(h)+\left[b_4h^4+b_6h^6+\cdots\right]\tag{2-6}\f] + +与第 1 步的外推过程类似,以\f$\frac h2\f$替换\f$\fml{2-6}\f$中的\f$h\f$,可得 + +\f[L=\psi\left(\frac h2\right)+\left[b_4\frac{h^4}{16}+b_6\frac{h^6}{64}+\cdots\right]\tag{2-7}\f] + +由\f$4^2\times\fml{2-7}-\fml{2-6}\f$可以得到 + +\f[L=\left[\frac{16}{15}\psi\left(\frac h2\right)-\frac1{15}\psi(h)\right]-\left[b_6\frac{h^6}{20}+\cdots\right]\tag{2-8}\f] + +此式表达了外推过程的第 2 步,说明\f$\psi(h)\f$和\f$\psi\left(\frac h2\right)\f$的一个线性组合提供了导数即\f$L\f$的新的计算公式,其精度提高至了\f$o\left(h^6\right)\f$ + +如果我们令 + +\f[\theta(h)=\frac{16}{15}\psi\left(\frac h2\right)-\frac1{15}\psi(h)\tag{2-9}\f] + +不难猜出,新的导数计算公式为 + +\f[L\approx\frac{64}{63}\theta\left(\frac h2\right)-\frac1{63}\theta(h)\tag{2-10}\f] + +读者可以自行验证,按照这个思路可以进一步外推,可执行任意多步得到精度不断提高的新公式,这就是 Richardson 外推原理。 + +#### 2.2 外推算法总结 + +令\f$\varphi(h)=\frac{f(x+h)-f(x-h)}{2h}\f$,外推\f$M\f$步,则外推算法的步骤如下 + +① 选取一个适当的\f$h\f$值,计算\f$M+1\f$个数,记为\f$T(*,*)\f$ + +\f[T(n,0)=\varphi\left(\frac h{2^n}\right),\qquad n=0,1,2,\cdots,M\tag{2-11}\f] + +② 按公式计算 + +\f[T(n,k)=\frac1{4^k-1}\left[4^kT(n,k-1)-T(n-1,k-1)\right],\qquad\left\{\begin{align} +k&=1,2,\cdots,M\\n&=k,k+1,\cdots,M +\end{align}\right.\tag{2-12}\f] + +按照以上步骤计算得到的\f$T(n,k)\f$,满足等式 + +\f[L=T(n,k)+o(h^{2k+2})\qquad(h\to0)\tag{2-13}\f] + +即\f$T(n,k)\f$具备\f$2k+2\f$阶的精度。 + +### 3. 如何使用 + +RMVL 中提供了一元函数以及多元函数的微分工具,求解一元函数导数时可使用 rm::derivative ,求解多元函数梯度是需要使用 rm::grad 。 + +以下展示了使用自动求导、数值微分的例子。 + +#### 3.1 创建项目 + +1. 添加源文件 `main.cpp` + ```cpp + #include + #include + + // 自定义函数 f(x)=x²+4x-3 + inline double quadratic(double x) { return x * x + 4 * x - 3; } + + int main() + { + double dydx = rm::derivative(quadratic, 1); + printf("f'(1) = %f\n", dydx); + } + ``` + +2. 添加 `CMakeLists.txt` + ```cmake + cmake_minimum_required(VERSION 3.10) + project(DerivativeDemo) + find_package(RMVL COMPONENTS core REQUIRED) + add_executable(demo main.cpp) + target_link_libraries(demo PRIVATE rmvl_core) + ``` + +#### 3.2 构建、运行 + +在项目根目录打开终端,输入 + +```bash +mkdir build +cd build +cmake .. +make -j2 +./demo +``` + +可以看到运行结果 + +``` +f'(1) = 6 +``` diff --git a/doc/tutorials/modules/algorithm/ew_topsis.md b/doc/tutorials/modules/algorithm/ew_topsis.md index 26f78573..e999da75 100644 --- a/doc/tutorials/modules/algorithm/ew_topsis.md +++ b/doc/tutorials/modules/algorithm/ew_topsis.md @@ -4,7 +4,7 @@ @author RoboMaster Vision Community @date 2023/01/11 -@prev_tutorial{tutorial_modules_runge_kutta} +@prev_tutorial{tutorial_modules_fminunc} @next_tutorial{tutorial_modules_kalman} diff --git a/doc/tutorials/modules/algorithm/fminbnd.md b/doc/tutorials/modules/algorithm/fminbnd.md new file mode 100644 index 00000000..4b1be81f --- /dev/null +++ b/doc/tutorials/modules/algorithm/fminbnd.md @@ -0,0 +1,158 @@ +一维最优化方法 {#tutorial_modules_fminbnd} +============ + +@author 赵曦 +@date 2024/05/06 +@version 1.0 +@brief 包含区间搜索以及黄金分割法的介绍 + +@prev_tutorial{tutorial_modules_auto_differential} + +@next_tutorial{tutorial_modules_fminunc} + +@tableofcontents + +------ + +\f[ +\def\red#1{\color{red}{#1}} +\def\teal#1{\color{teal}{#1}} +\def\green#1{\color{green}{#1}} +\def\transparent#1{\color{transparent}{#1}} +\def\orange#1{\color{orange}{#1}} +\def\fml#1{\text{(#1)}} +\f] + +### 1. 区间搜索 + +在使用一维优化方法搜索目标函数的极小值点时,首先要确定搜索区间\f$[a,b]\f$,这个搜索区间要包含目标函数的极小值点,而且是单峰区间,即在这个区间内,目标函数只有一个极小值点。 + +在单峰区间内,函数值具有“高—低—高”的特点。根据这一特点,可以采用进退法来找到搜索区间。 + +进退法分为 + +- 初始探察确定进退 +- 前进或者后退寻查 + +两步,其步骤如下 + +

+ +![图 1-1 向前一步](fminbnd_forward.png) + +![图 1-2 向前两步](fminbnd_forward.png) + +
+ +1. 选择一个初始点\f$\alpha_1\f$和一个初始步长\f$h\f$。 +2. 如图 1-1 所示,计算点\f$\alpha_1\f$和点\f$\alpha_1+h\f$对应的函数值\f$f(\alpha_1)\f$和\f$f(\alpha_1+h)\f$,令\f[f_1=f(\alpha_1),\quad f_2=f(\alpha_1+h)\f] +3. 比较\f$f_1\f$和\f$f_2\f$,若\f$f_1>f_2\f$,则执行前进运算,将步长加大\f$k\f$倍(如加大 2 倍),取新点\f$\alpha_1+3h\f$,如图 1-2 所示,计算其函数值,并令\f[f_1=f(\alpha_1+h),\quad f_2=f(\alpha_1+3h)\f]若\f$f_1< f_2\f$,则初始搜索区间端点为\f[a=\alpha_1,\quad b=\alpha_1+3h\f]若\f$f_1=f_2\f$,则初始搜索区间端点为\f[a=\alpha_1+h,\quad b=\alpha_1+3h\f]若\f$f_1>f_2\f$,则要继续做前进运算,且步长再加大两倍,取第 4 个点\f$\alpha_1+7h\f$,再比较第 3 和第 4 个点处的函数值……如此反复循环,直到在连续的 3 个点的函数值出现“两头大、中间小”的情况为止。 +4. 如果在 **步骤 3** 中出现\f$f_1< f_2\f$的情况,则执行后退运算,将步长变为负值,取新点\f$\alpha_1-h\f$,计算函数值,令\f[f_1=f(\alpha_1-h),\quad f_2=f(\alpha_1)\f]若\f$f_1>f_2\f$,则初始搜索区间端点为\f[a=\alpha_1-h,\quad b=\alpha_1+h\f]若\f$f_1=f_2\f$,则初始搜索区间端点为\f[a=\alpha_1-h,\quad b=\alpha_1\f]若\f$f_1< f_2\f$,则要继续做后退运算,且步长再加大两倍,取第 4 个点\f$\alpha_1-3h\f$,再比较第 3 和第 4 个点处的函数值……如此反复循环,直到在连续的 3 个点的函数值出现“两头大、中间小”的情况为止。 + +**示例** + +试用进退法确定目标函数\f$f(\alpha)=\alpha^2-5\alpha+8\f$的一维优化初始搜索区间\f$[a,b]\f$。设初始点\f$\alpha_1=0\f$,初始步长\f$h=1\f$ + +**解:** 由初始点\f$\alpha_1=0\f$,初始步长\f$h=1\f$,得\f[f_1=f(\alpha_1)=8,\quad f_2=f(\alpha_1+h)=4\f]因为\f$f_1>f_2\f$,所以执行前进运算,将步长加大 2 倍,取新点\f$\alpha_1+3h=3\f$,令\f[f_1=f(\alpha_1+h)=4,\quad f_2=f(\alpha_1+3h)=2\f]因为\f$f_1>f_2\f$,应继续做前进运算,且步长再加大 2 倍,取第 4 个点\f$\alpha_1+7h=7\f$。令\f[f_1=f(\alpha_1+3h)=2,\quad f_2=f(\alpha_1+7h)=22\f]此时\f$f_1< f_2\f$,在连续的 3 个点\f$\alpha_1+h\f$,\f$\alpha_1+3h\f$,\f$\alpha_1+7h\f$的函数值出现了“两头大、中间小”的情况,则初始搜索区间端点为\f[a=\alpha_1+h=1,\quad b=\alpha_1+7h=7\f] + +### 2. 黄金分割法 + +#### 2.1 区间消去 + +黄金分割法是利用区间消去法的原理,通过不断缩小单峰区间长度,即每次迭代都消去一部分不含极小值点的区间,使搜索区间不断缩小,从而逐渐逼近目标函数极小值点的一种优化方法。黄金分割法是直接寻优法,通过直接比较区间上点的函数值的大小来判断区间的取舍。这种方法具有计算简单、收敛速度快等优点。 + +如图1-2所示,在已确定的单峰区间[a,b]内任取\f$\alpha_1\f$,\f$\alpha_2\f$两点,计算并比较两点处的函数值\f$f(\alpha_1)\f$和\f$f(\alpha_2)\f$,可能出现3种情况: + +1. \f$f(\alpha_1)< f(\alpha_2)\f$,因为函数是单峰的,所以极小值点必定位于点\f$\alpha_2\f$的左侧,即\f$\alpha^*\in[a,\alpha_2]\f$,搜索区间可以缩小为\f$[a,\alpha_2]\f$,如图 2-1(a) 所示; +2. \f$f(\alpha_1)>f(\alpha_2)\f$,极小值点必定位于点\f$\alpha_1\f$的右侧,即\f$\alpha^*\in[\alpha_1,b]\f$,搜索区间可以缩小为\f$[\alpha_1,b]\f$,如图 2-1(b) 所示; +3. \f$f(\alpha_1)=f(\alpha_2)\f$,极小值点必定位于点\f$\alpha_1\f$和\f$\alpha_2\f$之间,即\f$\alpha^*\in[\alpha_1,\alpha_2]\f$,搜索区间可缩小为\f$[\alpha_1,\alpha_2]\f$,如图 2-1(c) 所示。 + +
+ +![图 2-1 搜索区间缩小示意图](region_reduce.png) + +
+ +根据上述方法,可在新搜索区间里再取两个新点比较函数值来继续缩小区间,但这样做效率较低,应该考虑利用已经计算过的区间内剩下的那个点。对于以上的 (1)、(2) 两种情况,可以在新搜索区间内取一点\f$x_3\f$和该区间内剩下的那个点(第 (1) 种情况的\f$\alpha_1\f$点或第 (2) 种情况的\f$\alpha_2\f$点)进行函数值的比较来继续缩短搜索区间。而第 (3) 种情况则要取两个新点进行比较,为统一起见,将前面 3 种情况综合为以下两种情况: + +1. 若\f$f(\alpha_1)< f(\alpha_2)\f$,则将搜索区间缩小为\f$[a,\alpha_2]\f$ +2. 若\f$f(\alpha_2)\ge f(\alpha_2)\f$,则将搜索区间缩小为\f$[\alpha_1,b]\f$。 + +#### 2.2 黄金分割法原理 + +黄金分割法就是基于上述原理来选择区间内计算点的位置的,它有以下要求: +1. 点\f$\alpha_1\f$和\f$\alpha_2\f$相对区间\f$[a,b]\f$的边界要对称布置,即区间\f$[a,\alpha_1)\f$的大小与区间\f$(\alpha_2,b]\f$的大小相等; +2. 每次计算一个新点,要求保留的区间长度\f$l\f$与原区间长度\f$L\f$之比等于被消去的区间长度\f$L-l\f$与保留区间长度\f$l\f$之比,即要求下式成立:\f[\frac lL=\frac{L-l}l\tag{2-1}\f] + +令\f[\lambda=\frac lL\f]将上式代入式\f$\fml{2-1}\f$,有\f[\lambda^2+\lambda-1=0\tag{2-2}\f]求解式\f$\fml{2-2}\f$,得\f[\lambda=\frac{\sqrt5-1}2\approx0.618\f] + +该方法保证每次迭代都以同一比率缩小区间,缩短率为\f$0.618\f$,故黄金分割法又称为\f$0.618\f$法。保留的区间长度为整个区间长度的\f$0.618\f$倍,消去的区间长度为整个区间长度的\f$0.382\f$倍。 + +黄金分割法的计算步骤如下: + +
+ +![图 2-2 黄金分割法原理](fminbnd_principle.png) + +
+ +1. 在\f$[a,b]\f$内取两点\f$\alpha_1\f$,\f$\alpha_2\f$,如图 2-2(a) 所示,使\f[\begin{align}\alpha_1&=a+0.382(b-a)\\\alpha_2&=a+0.618(b-a)\end{align}\f]令\f$f_1=f(\alpha_1)\f$,\f$f_2=f(\alpha_2)\f$。 +2. 比较\f$f_1\f$和\f$f_2\f$。当\f$f_1< f_2\f$时,消去区间\f$(\alpha_2,b]\f$。做置换\f$b=\alpha_2\f$,\f$\alpha_2=\alpha_1\f$并另取新点\f$\alpha_1=a+0.382(b-a)\f$,如图 2-2(b) 所示,令\f$f_1=f(\alpha_1)\f$。当\f$f_1\ge f_2\f$时,消去区间\f$[a,\alpha_1)\f$。做置换\f$a=\alpha_1\f$,\f$\alpha_1=\alpha_2\f$,\f$f_1=f_2\f$,并另取新点\f$\alpha_2=a+0.618(b-a)\f$,如图 2-2(c) 所示,令 \f$f_2=f(\alpha_2)\f$。 +3. 检查终止条件。若\f$b-a\le\epsilon\f$,则输出最优解\f$\alpha^*=\frac12(a+b)\f$和最优值\f$f(\alpha^*)\f$;否则转第 (2) 步。 + +### 3. 如何使用 + +RMVL 中提供了一维寻优的函数 rm::fminbnd ,以下展示了一维寻优的例子。 + +#### 3.1 创建项目 + +1. 添加源文件 `main.cpp` + ```cpp + #include + #include + + // 自定义函数 f(x)=x²+4x-3 + inline double quadratic(double x) { return x * x + 4 * x - 3; } + + int main() + { + // 确定搜索区间 + auto [x1, x2] = rm::region(quadratic, 0); + // 添加优化选项(容许误差和最大迭代次数) + rm::OptimalOptions options; + options.max_iter = 100; // 最大迭代次数是 100,不加以设置默认是 1000 + options.tol = 1e-4; // 容许误差是 1e-4,不加以设置默认是 1e-6 + // 一维寻优 + auto [x, fval] = rm::fminbnd(quadratic, x1, x2, options); + printf("x = %f\n", x); + printf("fval = %f\n", fval); + } + ``` + +2. 添加 `CMakeLists.txt` + ```cmake + cmake_minimum_required(VERSION 3.10) + project(FminbndDemo) + find_package(RMVL COMPONENTS core REQUIRED) + add_executable(demo main.cpp) + target_link_libraries(demo PRIVATE rmvl_core) + ``` + +#### 3.2 构建、运行 + +在项目根目录打开终端,输入 + +```bash +mkdir build +cd build +cmake .. +make -j2 +./demo +``` + +可以看到运行结果 + +``` +x = -2.000000 +fval = -7.000000 +``` \ No newline at end of file diff --git a/doc/tutorials/modules/algorithm/fminunc.md b/doc/tutorials/modules/algorithm/fminunc.md new file mode 100644 index 00000000..31ec157c --- /dev/null +++ b/doc/tutorials/modules/algorithm/fminunc.md @@ -0,0 +1,17 @@ +多维无约束最优化方法 {#tutorial_modules_fminunc} +============ + +@author 赵曦 +@date 2024/05/06 +@version 1.0 +@brief 包含最速下降、共轭梯度、Nelder-Mead 单纯形法的介绍 + +@prev_tutorial{tutorial_modules_fminbnd} + +@next_tutorial{tutorial_modules_ew_topsis} + +@tableofcontents + +------ + +@warning 未完成 \ No newline at end of file diff --git a/doc/tutorials/modules/algorithm/least_square.md b/doc/tutorials/modules/algorithm/least_square.md index ab5c7f12..97769e52 100644 --- a/doc/tutorials/modules/algorithm/least_square.md +++ b/doc/tutorials/modules/algorithm/least_square.md @@ -19,9 +19,9 @@ 最小二乘法诞生于统计学,其最初的目标是使用一条直线拟合一系列离散的点,那么我们该如何寻找这一直线?最小二乘法的原理是使得拟合直线的误差的平方和最小。这里的误差定义成每一个离散的点\f$x_i\f$与拟合直线的 **距离** 。在这一最初的用法中,点到直线的距离能够很直观的描述成点到直线的 **垂线** 的长度。
-![ls-line-vertical](ls-line-vertical.png) -图 1-1 +![图 1-1](ls-line-vertical.png) +
如图 1-1 所示,令直线外一点为\f$P\f$,直线为\f$l\f$,垂线为\f$l_0\f$,垂足为点\f$O\f$。最小二乘法就是构建一条直线,使得这些点\f$P_i\f$到这条直线的距离平方和最小,此时\f$O_i\f$点被称为 **最小二乘解** 。 @@ -57,9 +57,9 @@ 我们先用一个最简单的例子。
-![ls-vector-vertical](ls-vector-vertical.png) -图 1-2 +![图 1-2](ls-vector-vertical.png) +
对于图 1-2 中使用向量表示的方式,最小二乘解可以表示为向量\f$OP\f$的坐标,在这种情况下,\f$BP\f$与\f$OA\f$垂直,即: @@ -86,16 +86,14 @@ 这里 \f$A=\begin{bmatrix}2&1\\1&-1\\1&1\end{bmatrix}\f$,\f$\pmb b=(1,0,2)^T\f$,在图 1-3 中表示如下。
-![ls-equations](ls-equations.png) -图 1-3 +![图 1-3](ls-equations.png) +
一般的,对于一个系数矩阵 \f$A=(a_{ij})_{s\times{n}}\f$,\f$\pmb b=(b_1,b_2,\cdots,b_s)^T\f$,\f$\pmb x=(x_1,x_2,\cdots,x_s)^T\f$,若满足 \f$\text{rank}(A)\leq s\f$ 时,线性方程组没有数值解,但我们希望找到一组 **最优解** ,衡量此最优解的方法仍然可以采用最小二乘法。设法找出一组解 \f$\hat{\pmb x}=(x_1^0,\ x_2^0,\ x_3^0,\ \cdots,\ x_n^0)\f$ 使得每一项的误差 \f$\delta_i\f$ 平方和最小,如何定量这个误差?能否继续采用最小二乘法的点到直线的最短距离作为出发点?答案是肯定的,这里先给出误差平方和的表达式。 -\f[ -\delta^2=\sum_{i=0}^{s-1}(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n-b_i)^2\tag{2-2} -\f] +\f[\delta^2=\sum_{i=0}^{s-1}(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n-b_i)^2\tag{2-2}\f] 上式也可以写成 @@ -174,9 +172,9 @@ y&=x_1 这是一个参数方程,消去 \f$x_1\f$ 可以得到 \f$y=x\f$,这就是系数矩阵 \f$A=(1,\ 1)^T\f$ 所生成的列空间。
-![ls-eg1](ls-eg1.png) -图 2-1 +![图 2-1](ls-eg1.png) +
相当于现在需要在 \f$y=x\f$ 上找到一个点,使得其到 \f$\pmb b\f$ 的距离最短,这是初等几何的内容,做垂线即可,最终得到交点的坐标 @@ -196,9 +194,9 @@ y&=x_1 列空间外向量(右端项):\f$\pmb b=(1,0,2)^T\f$
-![ls-eg2](ls-eg2.png) -图 2-2 +![图 2-2](ls-eg2.png) +
需要满足 \f$(\pmb y-\pmb b)\perp\alpha\f$,则需要分别满足 \f$(\pmb y-\pmb b)\perp\pmb\alpha_1\f$ 和 \f$(\pmb y-\pmb b)\perp\pmb\alpha_2\f$,下面使用几何法对最小二乘法的原理进行验证。 @@ -408,9 +406,9 @@ d_1=(\phi_1,f)&=\sum_{i=0}^3\phi_1(t_i)f(t_i)=\sum_{i=0}^3t_i\times f(t_i)\\&=0+ 解得:\f$\left\{\begin{align}a_0&=-0.5\\a_1&=0.8\end{align}\right.\f$,即拟合曲线为 \f$y=-0.5+0.8t\f$,使用数学绘图软件验证,结果正确。
-![ls-eg3](ls-eg3.png) -图 4-1 +![图 4-1](ls-eg3.png) +
### 5. 部署使用 diff --git a/doc/tutorials/modules/tools/opcua.md b/doc/tutorials/modules/tools/opcua.md index c2642bb3..00c05e81 100644 --- a/doc/tutorials/modules/tools/opcua.md +++ b/doc/tutorials/modules/tools/opcua.md @@ -35,9 +35,9 @@ OPC UA 的设计目标是建立一种通用的、独立于厂商和平台的通 #### 地址空间
-![opcua](opcua.svg) -图 1-1 OPC UA 地址空间模型 +![图 1-1 OPC UA 地址空间模型](opcua.svg) +
1. 对象类型节点 rm::ObjectType :提供对象的定义,即对象的抽象,与类相当,且子类可以继承父类的特征,方便模型的扩充。该节点包括对象的各种数据类型,数据的语义,以及控制方式。OPC UA 命名空间 `0` 中规定了多个基础的对象类型节点。如使用最广的 BaseObjectType(在 RMVL 中表示为 `rm::nodeBaseObjectType`),所有对象类型节点都需要继承该节点再进行扩充。在对具体设备建模的过程中,应该将设备组成的各部分分解为不同的对象分别建模,再用引用节点将各部分按照实际设备中的关系相关联,从而得到完整设备的对象类型节点。 diff --git a/doc/tutorials/modules/tutorial_modules.md b/doc/tutorials/modules/tutorial_modules.md index 15073309..e0117a32 100644 --- a/doc/tutorials/modules/tutorial_modules.md +++ b/doc/tutorials/modules/tutorial_modules.md @@ -33,6 +33,9 @@ - @subpage tutorial_modules_least_square - @subpage tutorial_modules_func_iteration - @subpage tutorial_modules_runge_kutta +- @subpage tutorial_modules_auto_differential +- @subpage tutorial_modules_fminbnd +- @subpage tutorial_modules_fminunc #### 数据与信号处理 diff --git a/modules/core/CMakeLists.txt b/modules/core/CMakeLists.txt index 2f2772f8..30215d24 100644 --- a/modules/core/CMakeLists.txt +++ b/modules/core/CMakeLists.txt @@ -39,6 +39,6 @@ if(BUILD_PERF_TESTS) rmvl_add_test( core Performance DEPENDS core - DEPEND_TESTS benchmark::benchmark_main + DEPEND_TESTS benchmark::benchmark_main opencv_video ) endif(BUILD_PERF_TESTS) diff --git a/modules/core/include/rmvl/core/math.hpp b/modules/core/include/rmvl/core/math.hpp index 696aa5db..3858f930 100644 --- a/modules/core/include/rmvl/core/math.hpp +++ b/modules/core/include/rmvl/core/math.hpp @@ -121,31 +121,31 @@ template constexpr Tp rad2deg(Tp rad) { return rad * static_cast(180) / static_cast(PI); } /** - * @brief Point类型转换为Matx类型 + * @brief Point 类型转换为 Matx 类型 * * @tparam Tp 数据类型 - * @param[in] point Point类型变量 - * @return Matx类型变量 + * @param[in] point Point 类型变量 + * @return Matx 类型变量 */ template constexpr cv::Matx point2matx(cv::Point3_ point) { return cv::Matx(point.x, point.y, point.z); } /** - * @brief Matx类型转换为Point类型 + * @brief Matx 类型转换为 Point 类型 * * @tparam Tp 数据类型 - * @param[in] matx Matx类型变量 - * @return Point类型变量 + * @param[in] matx Matx 类型变量 + * @return Point 类型变量 */ template constexpr cv::Point3_ matx2point(cv::Matx matx) { return cv::Point3_(matx(0), matx(1), matx(2)); } /** - * @brief Matx类型转换为Vec类型 + * @brief Matx 类型转换为 Vec 类型 * * @tparam Tp 数据类型 - * @param[in] matx Matx类型变量 - * @return Vec类型变量 + * @param[in] matx Matx 类型变量 + * @return Vec 类型变量 */ template constexpr cv::Vec matx2vec(cv::Matx matx) { return cv::Vec(matx(0), matx(1), matx(2)); } @@ -421,6 +421,156 @@ typename ForwardIterator::value_type calculateModeNum(ForwardIterator first, For ->first; } +/** + * @brief 使用 `std::vector` 表示的向量加法 + * + * @tparam T 数据类型 + * @param[in] vec1 向量 1 + * @param[in] vec2 向量 2 + * @return 和向量 + */ +template +inline std::vector operator+(const std::vector &vec1, const std::vector &vec2) +{ + std::vector retval(vec1.size()); + std::transform(vec1.cbegin(), vec1.cend(), vec2.cbegin(), retval.begin(), std::plus()); + return retval; +} + +/** + * @brief 使用 `std::vector` 表示的向量减法 + * + * @tparam T 数据类型 + * @param[in] vec1 向量 1 + * @param[in] vec2 向量 2 + * @return 差向量 + */ +template +inline std::vector operator-(const std::vector &vec1, const std::vector &vec2) +{ + std::vector retval(vec1.size()); + std::transform(vec1.cbegin(), vec1.cend(), vec2.cbegin(), retval.begin(), std::minus()); + return retval; +} + +/** + * @brief 使用 `std::vector` 表示的向量自加 + * + * @tparam T 数据类型 + * @param[in] vec1 向量 1 + * @param[in] vec2 向量 2 + * @return 和向量 + */ +template +inline std::vector &operator+=(std::vector &vec1, const std::vector &vec2) +{ + std::transform(vec1.cbegin(), vec1.cend(), vec2.cbegin(), vec1.begin(), std::plus()); + return vec1; +} + +/** + * @brief 使用 `std::vector` 表示的向量自减 + * + * @tparam T 数据类型 + * @param[in] vec1 向量 1 + * @param[in] vec2 向量 2 + * @return 差向量 + */ +template +inline std::vector &operator-=(std::vector &vec1, const std::vector &vec2) +{ + std::transform(vec1.cbegin(), vec1.cend(), vec2.cbegin(), vec1.begin(), std::minus()); + return vec1; +} + +/** + * @brief 使用 `std::vector` 表示的向量取反 + * + * @tparam T 数据类型 + * @param[in] vec 向量 + * @return 向量取反后的结果 + */ +template +inline std::vector operator-(const std::vector &vec) +{ + std::vector retval(vec.size()); + std::transform(vec.cbegin(), vec.cend(), retval.begin(), std::negate()); + return retval; +} + +/** + * @brief 使用 `std::vector` 表示的向量乘法(数乘) + * + * @tparam T 数据类型 + * @param[in] vec 向量 + * @param[in] val 数乘因子 + * @return 向量乘法(数乘) + */ +template +inline std::vector operator*(const std::vector &vec, T val) +{ + std::vector retval(vec.size()); + std::transform(vec.cbegin(), vec.cend(), retval.begin(), [val](const T &x) { return x * val; }); + return retval; +} + +/** + * @brief 使用 `std::vector` 表示的向量乘法(数乘) + * + * @tparam T 数据类型 + * @param[in] val 数乘因子 + * @param[in] vec 向量 + * @return 向量乘法(数乘) + */ +template +inline std::vector operator*(T val, const std::vector &vec) { return vec * val; } + +/** + * @brief 使用 `std::vector` 表示的向量乘法(数乘) + * + * @tparam T 数据类型 + * @param[in] vec 向量 + * @param[in] val 数乘因子 + * @return 向量乘法(数乘) + */ +template +inline std::vector &operator*=(std::vector &vec, T val) +{ + std::transform(vec.cbegin(), vec.cend(), vec.begin(), [val](const T &x) { return x * val; }); + return vec; +} + +/** + * @brief 使用 `std::vector` 表示的向量除法(数乘) + * + * @tparam T 数据类型 + * @param[in] vec 向量 + * @param[in] val 除数 + * @return 向量除法(数乘) + */ +template +inline std::vector operator/(const std::vector &vec, T val) +{ + std::vector retval(vec.size()); + std::transform(vec.cbegin(), vec.cend(), retval.begin(), [val](const T &x) { return x / val; }); + return retval; +} + +/** + * @brief 使用 `std::vector` 表示的向量除法(数乘) + * + * @tparam T 数据类型 + * @param[in] val 除数 + * @param[in] vec 向量 + * @return 向量除法(数乘) + */ +template +inline std::vector &operator/=(std::vector &vec, T val) +{ + std::transform(vec.cbegin(), vec.cend(), vec.begin(), [val](const T &x) { return x / val; }); + return vec; +} + //! @} core } // namespace rm diff --git a/modules/core/include/rmvl/core/numcal.hpp b/modules/core/include/rmvl/core/numcal.hpp index dd2ffe8d..83188bcf 100644 --- a/modules/core/include/rmvl/core/numcal.hpp +++ b/modules/core/include/rmvl/core/numcal.hpp @@ -11,11 +11,11 @@ #pragma once -#include -#include #include #include #include +#include +#include //! @addtogroup core //! @{ @@ -23,6 +23,10 @@ //! @{ //! @brief 包含函数插值、曲线拟合、非线性方程(组)数值解、常微分方程数值解等数值计算算法 //! @} core_numcal +//! @defgroup core_optimal 最优化算法库 +//! @{ +//! @brief 包含一维函数最小值搜索、无约束多维函数最小值搜索等最优化算法 +//! @} core_optimal //! @} core namespace rm @@ -59,12 +63,7 @@ class Polynomial * @param[in] x 指定点的 x 坐标 * @return 多项式在指定点的函数值 */ - inline double operator()(double x) const noexcept - { - double y = _coeffs.back(); - std::for_each(_coeffs.rbegin() + 1, _coeffs.rend(), [&](double val) { y = y * x + val; }); - return y; - } + double operator()(double x) const noexcept; }; ///////////////////// 函数插值 ///////////////////// @@ -305,4 +304,92 @@ class RungeKutta4 : public RungeKutta //! @} core_numcal +//! @addtogroup core_optimal +//! @{ + +//! 一维函数 +using Func1d = std::function; + +//! 多维函数 +using FuncNd = std::function &)>; + +//! 梯度/导数计算模式 +enum DiffMode : uint8_t +{ + Diff_Central, //!< 中心差商 + Diff_Ridders, //!< Richardson 外推 +}; + +//! 多维函数最优化模式 +enum OptimizeMode : uint8_t +{ + Optm_ConjGrad, //!< 共轭梯度法 + Optm_Simplex, //!< 单纯形法 +}; + +//! 优化选项 +struct OptimalOptions +{ + DiffMode diff_mode{}; //!< 梯度计算模式,默认为中心差商 `Central` + OptimizeMode optm_mode{}; //!< 优化模式,默认为共轭梯度法 `ConjGrad` + int max_iter{1000}; //!< 最大迭代次数 + double dx{1e-2}; //!< 求解步长 + double tol{1e-6}; //!< 误差容限 +}; + +/** + * @brief 计算一元函数的导数 + * + * @param[in] func 多元函数 + * @param[in] x 指定位置的自变量 + * @param[in] mode 导数计算模式,默认为中心差商 `Diff_Central` + * @param[in] dx 坐标的微小增量,默认为 `1e-3` + * @return 函数在指定点的导数 + */ +double derivative(Func1d func, double x, DiffMode mode = Diff_Central, double dx = 1e-3); + +/** + * @brief 计算多元函数的梯度 + * + * @param[in] func 多元函数 + * @param[in] x 指定位置的自变量 + * @param[in] mode 梯度计算模式,默认为中心差商 `Diff_Central` + * @param[in] dx 计算偏导数时,坐标的微小增量,默认为 `1e-3` + * @return 函数在指定点的梯度向量 + */ +std::vector grad(FuncNd func, const std::vector &x, DiffMode mode = Diff_Central, double dx = 1e-3); + +/** + * @brief 采用进退法确定搜索区间 + * + * @param[in] func 一维函数 + * @param[in] x0 初始点 + * @param[in] delta 搜索步长 + * @return 搜索区间 + */ +std::pair region(Func1d func, double x0, double delta = 1); + +/** + * @brief 一维函数最小值搜索 + * + * @param[in] func 一维函数 + * @param[in] x1 搜索区间左端点 + * @param[in] x2 搜索区间右端点 + * @param[in] options 优化选项,默认为 `1e-4` 的误差容限和 `100` 的最大迭代次数 + * @return `[x, fval]` 最小值点和最小值 + */ +std::pair fminbnd(Func1d func, double x1, double x2, const OptimalOptions &options = {}); + +/** + * @brief 无约束多维函数最小值搜索 \cite ConjGrad \cite NelderMead + * + * @param[in] func 多维函数 + * @param[in] x0 初始点 + * @param[in] options 优化选项,默认为 `1e-4` 的误差容限和 `100` 的最大迭代次数 + * @return `[x, fval]` 最小值点和最小值 + */ +std::pair, double> fminunc(FuncNd func, const std::vector &x0, const OptimalOptions &options = {}); + +//! @} core_optimal + } // namespace rm diff --git a/modules/core/perf/perf_kalman.cpp b/modules/core/perf/perf_kalman.cpp index 72ade041..8e98bdbb 100644 --- a/modules/core/perf/perf_kalman.cpp +++ b/modules/core/perf/perf_kalman.cpp @@ -10,11 +10,10 @@ */ #include - +#include #include #include "rmvl/core/kalman.hpp" -#include namespace rm_test { @@ -76,7 +75,7 @@ void kalman42_opencv(benchmark::State &state) } } -BENCHMARK(kalman42_rmvl)->Name("KalmanFilter42 - RMVL")->Iterations(10); -BENCHMARK(kalman42_opencv)->Name("KalmanFilter42 - OpenCV")->Iterations(10); +BENCHMARK(kalman42_rmvl)->Name("kf (x_dim: 4, z_dim: 2) - by rmvl ")->Iterations(10); +BENCHMARK(kalman42_opencv)->Name("kf (x_dim: 4, z_dim: 2) - by opencv")->Iterations(10); } // namespace rm_test diff --git a/modules/core/perf/perf_optimal.cpp b/modules/core/perf/perf_optimal.cpp new file mode 100644 index 00000000..59732f98 --- /dev/null +++ b/modules/core/perf/perf_optimal.cpp @@ -0,0 +1,105 @@ +/** + * @file perf_optimal.cpp + * @author zhaoxi (535394140@qq.com) + * @brief 最优化算法库基准测试 + * @version 1.0 + * @date 2024-05-04 + * + * @copyright Copyright 2024 (c), zhaoxi + * + */ + +#include +#include + +#include "rmvl/core/numcal.hpp" + +namespace rm_test +{ + +/////////////////////// Quadratic Function /////////////////////// + +static double quadraticFunc(const std::vector &x) +{ + const auto &x1 = x[0], &x2 = x[1]; + return 60 - 10 * x1 - 4 * x2 + x1 * x1 + x2 * x2 - x1 * x2; +}; + +static void cg_quadratic_rmvl(benchmark::State &state) +{ + for (auto _ : state) + rm::fminunc(quadraticFunc, {0, 0}); +} + +class Quadratic : public cv::MinProblemSolver::Function +{ +public: + int getDims() const override { return 2; } + + double calc(const double *x) const override + { + const auto &x1 = x[0], &x2 = x[1]; + return 60 - 10 * x1 - 4 * x2 + x1 * x1 + x2 * x2 - x1 * x2; + } +}; + +static void cg_quadratic_cv(benchmark::State &state) +{ + for (auto _ : state) + { + auto quadratic = cv::makePtr(); + auto solver = cv::ConjGradSolver::create(quadratic); + cv::Vec2d x; + solver->minimize(x); + } +} + +BENCHMARK(cg_quadratic_rmvl)->Name("fminunc (conj_grad, quadratic) - by rmvl ")->Iterations(50); +BENCHMARK(cg_quadratic_cv)->Name("fminunc (conj_grad, quadratic) - by opencv")->Iterations(50); + +/////////////////////// Rosenbrock Function /////////////////////// + +static double rosenbrockFunc(const std::vector &x) +{ + const auto &x1 = x[0], &x2 = x[1]; + return 100 * (x2 - x1 * x1) * (x2 - x1 * x1) + (1 - x1) * (1 - x1); +}; + +void splx_rosenbrock_rmvl(benchmark::State &state) +{ + for (auto _ : state) + { + rm::OptimalOptions options; + options.optm_mode = rm::Optm_Simplex; + rm::fminunc(rosenbrockFunc, {5, 3}, options); + } +} + +class Rosenbrock : public cv::DownhillSolver::Function +{ +public: + int getDims() const override { return 2; } + + double calc(const double *x) const override + { + const auto &x1 = x[0], &x2 = x[1]; + return 100 * (x2 - x1 * x1) * (x2 - x1 * x1) + (1 - x1) * (1 - x1); + } +}; + +void splx_rosenbrock_cv(benchmark::State &state) +{ + for (auto _ : state) + { + auto rosenbrock = cv::makePtr(); + auto solver = cv::DownhillSolver::create(rosenbrock); + cv::Vec2d x(5, 3); + solver->setInitStep(x); + solver->minimize(x); + } +} + +BENCHMARK(splx_rosenbrock_rmvl)->Name("fminunc (simplex, rosenbrock) - by rmvl ")->Iterations(50); +BENCHMARK(splx_rosenbrock_cv)->Name("fminunc (simplex, rosenbrock) - by opencv")->Iterations(50); + +} // namespace rm_test diff --git a/modules/core/src/numcal.cpp b/modules/core/src/numcal.cpp index e90dd90b..11945799 100644 --- a/modules/core/src/numcal.cpp +++ b/modules/core/src/numcal.cpp @@ -9,10 +9,9 @@ * */ -#include - #include +#include "rmvl/core/math.hpp" #include "rmvl/core/numcal.hpp" #include "rmvl/core/util.hpp" @@ -21,6 +20,13 @@ namespace rm { +double Polynomial::operator()(double x) const noexcept +{ + double y = _coeffs.back(); + std::for_each(_coeffs.rbegin() + 1, _coeffs.rend(), [&](double val) { y = y * x + val; }); + return y; +} + ///////////////////// 函数插值 ///////////////////// Interpolator::Interpolator(const std::vector &xs, const std::vector &ys) @@ -139,36 +145,6 @@ RungeKutta::RungeKutta( RMVL_Error(RMVL_StsBadArg, "\"r[i].size()\" must be greater than or equal to the \"i\"."); } -// 加法 -template -inline static std::vector operator+(const std::vector &vec1, const std::vector &vec2) -{ - std::vector retval(vec1.size()); - std::transform(vec1.cbegin(), vec1.cend(), vec2.cbegin(), retval.begin(), std::plus()); - return retval; -} - -// 自加 -template -inline static std::vector &operator+=(std::vector &vec1, const std::vector &vec2) -{ - std::transform(vec1.cbegin(), vec1.cend(), vec2.cbegin(), vec1.begin(), std::plus()); - return vec1; -} - -// 数乘 -template -inline static std::vector operator*(const std::vector &vec, T val) -{ - std::vector retval(vec.size()); - std::transform(vec.cbegin(), vec.cend(), retval.begin(), [val](T x) { return x * val; }); - return retval; -} - -// 数乘 -template -inline static std::vector operator*(T val, const std::vector &vec) { return vec * val; } - /** * @brief 计算数值解 * diff --git a/modules/core/src/optimal.cpp b/modules/core/src/optimal.cpp new file mode 100644 index 00000000..cb08a3bf --- /dev/null +++ b/modules/core/src/optimal.cpp @@ -0,0 +1,286 @@ +/** + * @file optimal.cpp + * @author zhaoxi (535394140@qq.com) + * @brief 最优化算法库 + * @version 1.0 + * @date 2024-05-02 + * + * @copyright Copyright 2024 (c), zhaoxi + * + */ + +#include + +#include + +#include "rmvl/core/math.hpp" +#include "rmvl/core/numcal.hpp" + +namespace rm +{ + +// 中心差商计算一元函数导数 +static inline double partial(Func1d func, double x_dx, double dx) +{ + x_dx += dx; + double f1 = func(x_dx); + x_dx -= 2 * dx; + double f2 = func(x_dx); + x_dx += dx; + return (f1 - f2) / (2 * dx); +} + +// 中心差商计算多元函数偏导数 +static inline double partial(FuncNd func, std::vector &x_dx, std::size_t idx, double dx) +{ + x_dx[idx] += dx; + double f1 = func(x_dx); + x_dx[idx] -= 2 * dx; + double f2 = func(x_dx); + x_dx[idx] += dx; + return (f1 - f2) / (2 * dx); +} + +double derivative(Func1d func, double x, DiffMode mode, double dx) +{ + double x_dx{x}; + if (mode == Diff_Ridders) + { + double T00{partial(func, x_dx, 2 * dx)}; + double T10{partial(func, x_dx, dx)}; + double T20{partial(func, x_dx, dx / 2)}; + double T11{(4. * T10 - T00) / 3.}; + double T21{(4. * T20 - T10) / 3.}; + return (16. * T21 - T11) / 15.; + } + else + return partial(func, x_dx, dx); +} + +/** + * @brief 计算多元函数的梯度(无返回值) + * + * @param[in] func 多元函数 + * @param[in] x 指定位置的自变量 + * @param[out] xgrad 函数在指定点的梯度向量 + * @param[in] mode 梯度计算模式 + * @param[in] dx 计算偏导数时的步长 + * @return 函数在指定点的梯度向量 + */ +static void calcGrad(FuncNd func, const std::vector &x, std::vector &xgrad, DiffMode mode, double dx) +{ + std::vector x_dx{x}; + if (mode == Diff_Ridders) + for (std::size_t i = 0; i < x_dx.size(); ++i) + { + double T00{partial(func, x_dx, i, 2 * dx)}; + double T10{partial(func, x_dx, i, dx)}; + double T20{partial(func, x_dx, i, dx / 2)}; + double T11{(4. * T10 - T00) / 3.}; + double T21{(4. * T20 - T10) / 3.}; + xgrad[i] = (16. * T21 - T11) / 15.; + } + else + for (std::size_t i = 0; i < x_dx.size(); ++i) + xgrad[i] = partial(func, x_dx, i, dx); +} + +std::vector grad(FuncNd func, const std::vector &x, DiffMode mode, double dx) +{ + std::vector ret(x.size()); + calcGrad(func, x, ret, mode, dx); + return ret; +} + +/** + * @brief 计算向量的二范数 + * + * @param[in] x 向量 + * @return 二范数 + */ +static inline double norm(const std::vector &x) +{ + double retval{}; + for (auto &&v : x) + retval += v * v; + return std::sqrt(retval); +} + +std::pair region(Func1d func, double x0, double delta) +{ + double f1{func(x0)}, f2{func(x0 + delta)}; + if (f1 > f2) + { + f1 = f2; + f2 = func(x0 + 3 * delta); + return f1 < f2 ? std::make_pair(x0, x0 + 3 * delta) : (f1 == f2 ? std::make_pair(x0 + delta, x0 + 3 * delta) : region(func, x0 + delta, 2 * delta)); + } + else if (f1 < f2) + { + f2 = f1; + f1 = func(x0 - delta); + return f1 > f2 ? std::make_pair(x0 - delta, x0 + delta) : (f1 == f2 ? std::make_pair(x0 - delta, x0) : region(func, x0 - 3 * delta, 2 * delta)); + } + else + return {x0, x0 + delta}; +} + +std::pair fminbnd(Func1d func, double x1, double x2, const OptimalOptions &options) +{ + constexpr double phi = 0.618033988749895; + double a1{x1 + (1.0 - phi) * (x2 - x1)}, a2{x1 + phi * (x2 - x1)}; + double f1{func(a1)}, f2{func(a2)}; + for (int i = 0; i < options.max_iter; ++i) + { + if (std::abs(x1 - x2) < options.tol) + break; + if (f1 < f2) // 替换右端点 + { + x2 = a2, a2 = a1, f2 = f1; + a1 = x1 + (1.0 - phi) * (x2 - x1); + f1 = func(a1); + } + else // 替换左端点 + { + x1 = a1, a1 = a2, f1 = f2; + a2 = x1 + phi * (x2 - x1); + f2 = func(a2); + } + } + return {0.5 * (x1 + x2), func(0.5 * (x1 + x2))}; +} + +// 共轭梯度法 +static double fminunc_cg(FuncNd func, std::vector &xk, const OptimalOptions &options) +{ + std::vector s = -xk; + std::vector xk_grad = grad(func, xk, options.diff_mode, options.dx), xk2_grad(xk_grad.size()); + // 判断是否收敛 + double nbl_xk = norm(xk_grad); + if (nbl_xk < options.tol) + return func(xk); + // 一维搜索函数 + auto func_alpha = [&](double alpha) { return func(xk + alpha * s); }; + double retfval{}; + for (int i = 0; i < options.max_iter; ++i) + { + // 一维搜索 alpha + auto [a, b] = region(func_alpha, 1); + RMVL_Assert(b > 0); + auto [alpha, fval] = fminbnd(func_alpha, a < 0. ? 0. : a, b, options); + // 更新 xk,并计算对应的梯度 + for (std::size_t j = 0; j < xk.size(); ++j) + xk[j] += alpha * s[j]; + retfval = fval; + calcGrad(func, xk, xk2_grad, options.diff_mode, options.dx); + auto nbl_xk2 = norm(xk2_grad); + if (nbl_xk2 < options.tol) + break; + // 计算 beta + double nbl_xk_xk2 = std::inner_product(xk_grad.begin(), xk_grad.end(), xk2_grad.begin(), 0.0); + double beta = std::max((nbl_xk2 * nbl_xk2 - nbl_xk_xk2) / (nbl_xk * nbl_xk), 0.0); + // 更新搜索方向 + for (std::size_t j = 0; j < s.size(); ++j) + s[j] = -xk2_grad[j] + beta * s[j]; + xk_grad = xk2_grad; + } + return retfval; +} + +// 单纯形法 +static double fminunc_splx(FuncNd func, std::vector &xk, const OptimalOptions &options) +{ + const std::size_t dim = xk.size(); // 维度 + const std::size_t N = dim + 1; // 单纯形点的个数 + std::vector, double>> splx(N); + for (auto &p : splx) + p = {xk, 0}; + for (std::size_t i = 0; i < dim; ++i) + splx[i + 1].first[i] += 100 * options.dx; + for (std::size_t i = 0; i < N; ++i) + splx[i].second = func(splx[i].first); + // 单纯形迭代 + for (int i = 0; i < options.max_iter; ++i) + { + std::sort(splx.begin(), splx.end(), [](const auto &a, const auto &b) { return a.second < b.second; }); + // 求出除最大值点外的所有点的中心 + std::vector xc(dim); + std::for_each(splx.begin(), splx.end() - 1, [&](const auto &vp) { xc += vp.first; }); + xc /= static_cast(N - 1); + // 反射 xr = xc + alpha * (xc - xh),其中 alpha = 1 + std::vector xr(dim); + for (std::size_t j = 0; j < dim; ++j) + xr[j] = 2 * xc[j] - splx.back().first[j]; + + double fxr = func(xr); + // f(xn-1) <= f(xr),反射点函数值大于最差点,则重新计算反射点(反压缩) + if (splx.back().second <= fxr) + { + // 压缩 xs = xc - beta * (xc - xh),其中 beta = 0.5 + xr = xc + splx.back().first; + xr /= 2.0; + fxr = func(xr); + } + + // f(xr) < f(x0),反射点函数值小于最优点 + if (fxr < splx[0].second) + { + // 扩展 xe = xc + gamma * (xc - xh),其中 gamma = 2 + std::vector xe(dim); + for (std::size_t j = 0; j < dim; ++j) + xe[j] = 3 * xc[j] - 2 * splx.back().first[j]; + double fxe = func(xe); + splx.back() = fxe < fxr ? std::make_pair(xe, fxe) : std::make_pair(xr, fxr); + } + // f(x0) <= f(xr) < f(xn-2),反射点函数值大于最优点,小于次差点 + else if (splx[0].second <= fxr && fxr < splx[N - 2].second) + splx.back() = {xr, fxr}; + // f(xn-2) <= f(xr) < f(xn),反射点函数值大于次差点,小于最差点 + else + { + // 压缩 xs = xc + beta * (xc - xh),其中 beta = 0.5 + std::vector xs(dim); + for (std::size_t j = 0; j < dim; ++j) + xs[j] = 1.5 * xc[j] - 0.5 * splx.back().first[j]; + double fxs = func(xs); + if (fxs < splx.back().second) + splx.back() = {xs, fxs}; + else + { + for (std::size_t i = 1; i < N; ++i) + { + for (std::size_t j = 0; j < dim; ++j) + splx[i].first[j] = 0.5 * (splx[i].first[j] + splx[0].first[j]); + splx[i].second = func(splx[i].first); + } + } + } + // 判断是否收敛 + if (std::abs(splx.back().second - splx.front().second) < options.tol) + break; + } + xk = std::move(splx[0].first); + return splx[0].second; +} + +std::pair, double> fminunc(FuncNd func, const std::vector &x0, const OptimalOptions &options) +{ + if (x0.empty()) + RMVL_Error(RMVL_StsBadArg, "x0 is empty"); + std::vector xk{x0}; + double fval{}; + switch (options.optm_mode) + { + case Optm_ConjGrad: + fval = fminunc_cg(func, xk, options); + break; + case Optm_Simplex: + fval = fminunc_splx(func, xk, options); + break; + default: + break; + } + return {xk, fval}; +} + +} // namespace rm diff --git a/modules/core/test/test_optimal.cpp b/modules/core/test/test_optimal.cpp new file mode 100644 index 00000000..0ee7af9a --- /dev/null +++ b/modules/core/test/test_optimal.cpp @@ -0,0 +1,70 @@ +/** + * @file test_optimal.cpp + * @author zhaoxi (535394140@qq.com) + * @brief 最优化算法库测试 + * @version 1.0 + * @date 2024-05-04 + * + * @copyright Copyright 2024 (c), zhaoxi + * + */ + +#include +#include + +#include + +namespace rm_test +{ + +static inline double f1d(double x) { return x * x - 4 * x + 7; } +static inline double rosenbrock(const std::vector &x) +{ + const auto &x1 = x[0], &x2 = x[1]; + return 100 * (x2 - x1 * x1) * (x2 - x1 * x1) + (1 - x1) * (1 - x1); +} +static inline double quadratic(const std::vector &x) +{ + const auto &x1 = x[0], &x2 = x[1]; + return 60 - 10 * x1 - 4 * x2 + x1 * x1 + x2 * x2 - x1 * x2; +} + +TEST(Optimal, derivative) +{ + auto func = [](double x) { return std::pow(x, 7) - std::pow(x, 5); }; + auto val_central = rm::derivative(func, 1); + EXPECT_NEAR(val_central, 2, 1e-3); + auto val_ridders = rm::derivative(func, 1, rm::Diff_Ridders); + EXPECT_NEAR(val_ridders, 2, 1e-12); +} + +TEST(Optimal, fminbnd) +{ + auto [x1, x2] = rm::region(f1d, -5); + EXPECT_LE(x1, 2); + EXPECT_GE(x2, 2); + auto [x, fval] = rm::fminbnd(f1d, x1, x2); + EXPECT_NEAR(x, 2, 1e-4); + EXPECT_NEAR(fval, 3, 1e-4); +} + +TEST(Optimal, fminunc_simplex) +{ + rm::OptimalOptions options; + options.optm_mode = rm::Optm_Simplex; + auto [x, fval] = rm::fminunc(rosenbrock, {1, -2}, options); + EXPECT_NEAR(x[0], 1, 1e-2); + EXPECT_NEAR(x[1], 1, 1e-2); + EXPECT_NEAR(fval, 0, 1e-2); +} + +TEST(Optimal, fminunc_conjgrad) +{ + rm::OptimalOptions options; + auto [x, fval] = rm::fminunc(quadratic, {0, 0}, options); + EXPECT_NEAR(x[0], 8, 1e-4); + EXPECT_NEAR(x[1], 6, 1e-4); + EXPECT_NEAR(fval, 8, 1e-4); +} + +} // namespace rm_test