我想解决一个带有9个未知数的9个非线性方程组，并用Matlab求解。
所有9个未知数都作为多项式的比率耦合（参见myfun lower）
x02=[5000,5000,5000,0.4,0.4,0.4,0.4,0.4,0.4];
ctrl2=[9894+1i*0.118,9894+1i*0.118,9894+1i*0.118,0.5,0.5,0.5,0.5,0.5,0.5];
f2 = @(x) myfun(x,Peff);
options2 = optimoptions('fsolve','Algorithm','trust-region-dogleg','Display','iter-detailed'...
,'MaxFunEvals', 100000, 'MaxIter', 100000,'TolX',1e-12,'TolFun',1e-12,...
'Jacobian','on');
[x2,F2,exitflag2,output2] = fsolve(f2,x02,options2);
function [F,J] = myfun( x, p)
% System of equation
F(1) = -(x(1) - x(1)*x(8)*x(9))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(1,1);
F(2) = -(x(2)*x(4) + x(2)*x(6)*x(9))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(1,2);
F(3) = -(x(3)*x(6) + x(3)*x(4)*x(8))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(1,3);
F(4) = -(x(1)*x(5) + x(1)*x(8)*x(7))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(2,1);
F(5) = -(x(2) - x(2)*x(6)*x(7))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(2,2);
F(6) = -(x(3)*x(8) + x(3)*x(6)*x(5))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(2,3);
F(7) = -(x(1)*x(7) + x(1)*x(5)*x(9))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(3,1);
F(8) = -(x(2)*x(9) + x(2)*x(4)*x(7))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(3,2);
F(9) = -(x(3) - x(3)*x(4)*x(5))/(x(4)*x(5) + x(6)*x(7) + x(8)*x(9) + x(4)*x(8)*x(7) + x(6)*x(5)*x(9) - 1) - p(3,3);
%% Jacobian
I compute the Jacobian myself but will spare you the detail, as it is considerably long
end
Norm of First-order Trust-region
Iteration Func-count f(x) step optimality radius
0 1 2.45042e+19 1.39e+14 1
1 2 2.45042e+19 1 1.39e+14 1
2 3 2.45031e+19 0.25 4.77e+15 0.25
3 4 2.45031e+19 0.625 4.77e+15 0.625
4 5 2.45031e+19 0.15625 4.77e+15 0.156
5 6 2.44992e+19 0.0390625 6.8e+16 0.0391
6 7 2.44992e+19 0.0976562 6.8e+16 0.0977
7 8 2.44992e+19 0.0244141 6.8e+16 0.0244
8 9 2.4495e+19 0.00610352 2.03e+17 0.0061
9 10 2.4495e+19 0.0152588 2.03e+17 0.0153
10 11 2.4486e+19 0.0038147 7.67e+17 0.00381
11 12 2.4486e+19 0.00953674 7.67e+17 0.00954
12 13 2.44592e+19 0.00238419 4.62e+18 0.00238
13 14 2.44592e+19 0.00596046 4.62e+18 0.00596
14 15 2.40048e+19 0.00149012 5.62e+20 0.00149
15 16 2.40048e+19 0.00372529 5.62e+20 0.00373
16 17 2.40048e+19 0.000931323 5.62e+20 0.000931
17 18 2.40048e+19 0.000232831 5.62e+20 0.000233
18 19 2.36832e+19 5.82077e-05 1.52e+21 5.82e-05
19 20 2.36832e+19 0.000145519 1.52e+21 0.000146
20 21 2.3131e+19 3.63798e-05 4.24e+21 3.64e-05
21 22 2.3131e+19 9.09495e-05 4.24e+21 9.09e-05
22 23 2.21355e+19 2.27374e-05 1.26e+22 2.27e-05
23 24 2.21355e+19 5.68434e-05 1.26e+22 5.68e-05
24 25 2.01772e+19 1.42109e-05 4.2e+22 1.42e-05
25 26 2.01772e+19 3.55271e-05 4.2e+22 3.55e-05
26 27 1.5592e+19 8.88178e-06 1.76e+23 8.88e-06
27 28 1.5592e+19 2.22045e-05 1.76e+23 2.22e-05
28 29 1.17854e+18 5.55112e-06 7.24e+23 5.55e-06
29 30 3.43734e+16 1.38778e-05 1.9e+23 1.39e-05
30 31 1.23843e+15 3.46945e-05 4.04e+22 3.47e-05
31 32 1.23843e+15 8.67362e-05 4.04e+22 8.67e-05
32 33 7.49991e+13 2.1684e-05 4.25e+21 2.17e-05
33 34 7.49991e+13 5.42101e-05 4.25e+21 5.42e-05
34 35 3.19073e+13 1.35525e-05 3.46e+21 1.36e-05
35 36 3.19073e+13 3.38813e-05 3.46e+21 3.39e-05
36 37 3.19073e+13 8.47033e-06 3.46e+21 8.47e-06
37 38 3.19073e+13 2.11758e-06 3.46e+21 2.12e-06
38 39 3.19073e+13 5.29396e-07 3.46e+21 5.29e-07
39 40 3.19073e+13 1.32349e-07 3.46e+21 1.32e-07
40 41 3.19073e+13 3.30872e-08 3.46e+21 3.31e-08
41 42 3.19073e+13 8.27181e-09 3.46e+21 8.27e-09
42 43 3.19073e+13 2.06795e-09 3.46e+21 2.07e-09
43 44 3.19073e+13 5.16988e-10 3.46e+21 5.17e-10
44 45 3.16764e+13 1.29247e-10 3e+21 1.29e-10
45 46 3.16764e+13 1.29247e-10 3e+21 1.29e-10
46 47 3.16764e+13 3.23117e-11 3e+21 3.23e-11
47 48 3.16764e+13 8.07794e-12 3e+21 8.08e-12
48 49 3.16764e+13 2.01948e-12 3e+21 2.02e-12
49 50 3.16764e+13 5.04871e-13 3e+21 5.05e-13
50 51 3.16764e+13 1.26218e-13 3e+21 1.26e-13
51 52 3.16764e+13 3.15544e-14 3e+21 3.16e-14
52 53 3.16764e+13 7.88861e-15 3e+21 7.89e-15
53 54 3.16764e+13 1.97215e-15 3e+21 1.97e-15
54 55 3.16764e+13 4.93038e-16 3e+21 4.93e-16
fsolve stopped because the relative norm of the current step, 1.097484e-16, is less than
max(options.TolX^2,eps) = 2.220446e-16. However, the sum of squared function values,
r = 3.167644e+13, exceeds sqrt(options.TolFun) = 1.000000e-06.
Optimization Metric Options
relative norm(step) = 1.10e-16 max(TolX^2,eps) = 2e-16 (selected)
r = 3.17e+13 sqrt(TolFun) = 1.0e-06 (selected)
exitflag2 =
-2
output2 =
iterations: 54
funcCount: 55
algorithm: 'trust-region-dogleg'
firstorderopt: 3.000805388686251e+21
message: 'No solution found.…'
初步猜测
x02 =
1.0e+03 *
Columns 1 through 5
5.000000000000000 5.000000000000000 5.000000000000000 0.000400000000000 0.000400000000000
Columns 6 through 9
0.000400000000000 0.000400000000000 0.000400000000000 0.000400000000000
fsolve给出的解决方案
x2 =
1.0e+03 *
Columns 1 through 2
5.000098340971978 - 0.000000066639557i 5.000100855522207 + 0.000000027141142i
Columns 3 through 4
5.000100887684736 + 0.000000021333305i 0.000500827051867 + 0.000000033172152i
Columns 5 through 6
0.000498312570833 - 0.000000060511167i 0.000500859436647 + 0.000000027409553i
Columns 7 through 8
0.000500831092720 + 0.000000033506374i 0.000500831171443 + 0.000000033543065i
Column 9
0.000498337065684 - 0.000000066909835i
解决方案应该是什么（控制）
ctrl2 =
1.0e+03 *
Columns 1 through 2
9.894000000000000 + 0.355900000000000i 9.894000000000000 + 0.355900000000000i
Columns 3 through 4
9.894000000000000 + 0.355900000000000i 0.000499999000000 + 0.000000000000000i
Columns 5 through 6
0.000499999000000 + 0.000000000000000i 0.000499999000000 + 0.000000000000000i
Columns 7 through 8
0.000499999000000 + 0.000000000000000i 0.000499999000000 + 0.000000000000000i
Column 9
0.000499999000000 + 0.000000000000000i
正如你所看到的那样，在没有离初始猜测很远的情况下，解决崩溃的速度相当快。
我尝试更改TolX TolFun算法但仍然崩溃（根据Matlab网站的建议，在算法失败时该怎么做）
该算法在fsolve和lsqnonlin（levnberg，trust-region-dogleg，trust-region-reflective）中的其他算法崩溃类似
你能帮我弄清楚我需要做些什么来建立一个 将收敛到控制值的算法？
谢谢
答案 0 :(得分：0)
您遇到的问题实际上解算器没有任何问题。更有可能的是，你只需要在问题中使用许多局部最小化器，并且优化算法就会陷入困境。你真的没什么可做的......几乎所有你手上的优化求解器都是基于梯度下降的原理（或多或少），所以它们会引导你去一个最小化器，但不一定是全球性的。有些人做的是生成一大组随机初始条件，并为每个初始条件运行优化求解器。这并不能保证您获得的解决方案之一是全局最小化器，但至少您应该能够比单个初始猜测做得更好。
如果你真的需要找到全局最小化器，那么你需要使用某种类型的“Groebner-based”解决方法。这些类型的求解器基于“消除理论”，并将产生包含所有全局最小化器的单变量多项式。然而，这些解算器通常是符号的，极其耗费内存，并且基本上不能保证在有限的时间内完成。我怀疑（我真的不知道）这些问题是NP难的。根据我的经验，像你展示的那个看似很小的问题可能需要很长时间才能解决（如果有的话）。