如何在Matlab中为多个变量的超定二次系统找到argmin/最佳拟合/优化 [英] How to find argmin/best fit/optimize for an overdetermined quadratic system for multiple variables in Matlab

问题描述

我有100个方程式和5个变量. Matlab中有一个函数可以用来找到这些方程的最佳解吗?

I have 100 equations with 5 variables. Is there a function in Matlab which I can use to find the optimal solution of these equations?

我的问题是找到argmin ||(a-ic)^ 2 +(b-jd)^ 2 + e-h(i，j)||在所有i，j的范围从-10到10.即

My problem is to find argmin ||(a-ic)^2 + (b-jd)^2 + e - h(i,j)|| over all i, j from -10 to 10. ie.

%% Note: not Matlab code. Just showing the Math.
for i = -10:10
    for j = -10:10
        (a-ic)^2 + (b-jd)^2 + e = h(i,j)

已知:h(i,j)是10*10矩阵，而i,j是索引

known: h(i,j) is a 10*10 matrix,and i,j are indexes

预期:a,b,c,d,e

推荐答案

您可以尝试使用

You can try using lsqnonlin as follows.

%% define a helper function in your .m file
function f = fun(x)
   a=x(1); b=x(2); c=x(3); d=x(4); e=x(5); % Using variable names from your question. In other situations, be careful when overwriting e.
   f=zeros(21*21,0);  % size(f) is taken from your question. You should make this a variable for good practice.
   for i = -10:10
      for j = -10:10
         f(10*(i+10+1)+(j+10+1)) = (a-i*c)^2 + (b-j*d)^2 + e - h(i,j); % 10 is taken from your question. 
      end
   end
end

(此外，为什么您的h(i，j)取负指数?)

在主要功能中，您只需编写即可

In your main function you can simply write

function out=myproblem(x0)
    out=lsqnonlin(@fun,x0);
end

在您的cmd中，您可以使用特定的初始尝试进行呼叫，例如

In your cmd, you can call with specific initial try such as

myproblem([0,0,0,0,0])

助手功能优于匿名功能，因为根据我的经验，助手会被JIT加速，而匿名则不会.我还选择在循环中重塑形状，而不是在实际之后调用reshape，因为我希望reshape花费大量的额外时间.请记住，fun中的O(1)不是lsqnonlin中的O(1).

Helper function over anonymous because in my experience helpers get sped up by JIT while anonymous do not. I also opted to reshape in the loops as an opposed to actually call reshape after because I expect reshape to cost significant extra time. Remember that O(1) in fun is not O(1) in lsqnonlin.

(一如既往，不能保证解决非线性问题.)

(As always, a solution to a nonlinear problem is not guaranteed.)

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