MATLAB 中的优化符号计算

Question

所以这是一个结合符号计算和优化的问题。我有一个包含 phi21、phi31 和 phi32 的三个微分方程组。方程式中有四个参数我最终想优化 k1、k2、f 和 s。我在下面的代码中设置了方程式和雅可比行列式的构造：

syms phi21 phi31 phi32 k1 k2 f s a

w1 = (2*pi/24)*0.99;
w2 = (2*pi/24)*1.01;
w3 = (2*pi/24)*1.02;

f21 = (w2 - w1) + (1/3)*(-2*k1*sin(phi21) - 2*k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) + k1*sin(phi32) + k2*sin(2*phi32)) + 2*f*cos((2*s - phi21)/2)*sin(-phi21/2);
f31 = (w3 - w1) + (1/3)*(k1*sin(phi21) + k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) - 2*k1*sin(phi32) - 2*k2*sin(2*phi32)) + 2*f*cos((2*s - phi31)/2)*sin(-phi31/2);
f32 = (w3 - w2) + (1/3)*(-k1*sin(phi21) - k2*sin(2*phi21) - 2*k1*sin(phi31) - 2*k2*sin(2*phi31) - k1*sin(phi32) - k2*sin(2*phi32)) + + 2*f*cos((2*s - phi32)/2)*sin(-phi32/2);

df21d21 = diff(f21, phi21);
df21d31 = diff(f21, phi31);
df21d32 = diff(f21, phi32);

df31d21 = diff(f31, phi21);
df31d31 = diff(f31, phi31);
df31d32 = diff(f31, phi32);

df32d21 = diff(f32, phi21);
df32d31 = diff(f32, phi31);
df32d32 = diff(f32, phi32);


J = [df21d21 df21d31 df21d32; df31d21 df31d31 df31d32; df32d21 df32d31 df32d32];
lambda = eig(J);
rlambda = real(lambda);

srlambda = subs(rlambda, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]);
seq = [subs(f21, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f31, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f32, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271])];

完成此操作后，我希望进行优化以使 f21 = f31 = f32 = 0 并且特征值均为负数。但是，我不知道如何在某些非线性优化过程中使用我的符号表达式。我有一些代码如下所示：

x0 = [];
lb = [];
ub = [];

[sol, fval, exitflag, output] = fmincon(@eq1, x0, A, b, Aeq, beq, lb, ub, @constraints)

function objfun = eq1(k)
objfun = ;
end

function [c, ceq] = constraints(k)
c = [];
ceq = [];
end

我可以在其中为我的 f21、f31、f32 条件指定初始搜索点、上限和下限以及向量 ceq 和我的特征值条件的向量 c。我已经知道有几个问题。首先，优化部分需要 k(1)、k(2)、k(3) 和 k(4) 形式的变量，而不是 k1、k2、f，和 s。有没有办法轻松做到这一点？其次，我是否需要将符号约束转换为 MATLAB 函数？可能还有其他问题，但我不确定。任何帮助将不胜感激:)

Answer 1

您使用 matlabFunction 将所有需要的符号表达式（分别为 f21、f31、f32 或 seq）转换为可执行的 Matlab 函数.这将使它们可执行（并输出双精度而不是符号值），并让它们接受按字母顺序排序的多个输入参数。

因此，matlabFunction(seq) 将生成一个以 (f,k1,k2,s) 作为输入参数的匿名函数。您还可以使用 matlabFunction 的 'File' 参数将函数存储在文件中。

为了让这个函数接受一个你想要优化的参数向量，你可以写一个小的 'wrapper':

f_seq = matlabFunction(seq); % anonymous function that takes (f,k1,k2,s) as inputs
f_seq2 = @(p) f_seq(p(1),p(2),p(3),p(4)); % anonymous function that takes a 4 element vector

我建议将所有函数保存在文件中（也是包装器），因为对象函数有自己的工作区（即无法访问基础工作区中的匿名函数句柄）。

Answer 2

使用rinkert提到的matlabFunction转换syms 功能 function handle
等于f21 = f31 = f32等价于f21 - f31 = 0 and f21 - f32 = 0
要仅满足约束条件，请将 objective 函数定义为一个常数函数

eq = @(k)0

请仔细阅读评论

syms phi21 phi31 phi32 k1 k2 f s a

w1 = (2*pi/24)*0.99;
w2 = (2*pi/24)*1.01;
w3 = (2*pi/24)*1.02;

f21 = (w2 - w1) + (1/3)*(-2*k1*sin(phi21) - 2*k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) + k1*sin(phi32) + k2*sin(2*phi32)) + 2*f*cos((2*s - phi21)/2)*sin(-phi21/2);
f31 = (w3 - w1) + (1/3)*(k1*sin(phi21) + k2*sin(2*phi21) - k1*sin(phi31) - k2*sin(2*phi31) - 2*k1*sin(phi32) - 2*k2*sin(2*phi32)) + 2*f*cos((2*s - phi31)/2)*sin(-phi31/2);
f32 = (w3 - w2) + (1/3)*(-k1*sin(phi21) - k2*sin(2*phi21) - 2*k1*sin(phi31) - 2*k2*sin(2*phi31) - k1*sin(phi32) - k2*sin(2*phi32)) + + 2*f*cos((2*s - phi32)/2)*sin(-phi32/2);

df21d21 = diff(f21, phi21);
df21d31 = diff(f21, phi31);
df21d32 = diff(f21, phi32);

df31d21 = diff(f31, phi21);
df31d31 = diff(f31, phi31);
df31d32 = diff(f31, phi32);

df32d21 = diff(f32, phi21);
df32d31 = diff(f32, phi31);
df32d32 = diff(f32, phi32);


J = [df21d21 df21d31 df21d32; df31d21 df31d31 df31d32; df32d21 df32d31 df32d32];
lambda = eig(J);
rlambda = real(lambda);

srlambda = subs(rlambda, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]);
seq = [subs(f21, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f31, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271]), subs(f32, [phi21, phi31, phi32], [0.35475, 0.58305, 0.2271])];

% Transform syms function to function handles

f21 = matlabFunction(seq(1));
f31 = matlabFunction(seq(2));
f32 = matlabFunction(seq(3));

lambda = matlabFunction(srlambda);

% Inequality constraint, input is passed as a vector 
c =  @(k)lambda(k(1), k(2), k(3), k(4));

% Equality constraint, input is passed as a vector 
% f21 = f31 = f32 --> f21 -f31 = 0 and f21 -f32 = 0

ceq = @(k)[f21(k(1), k(2), k(3), k(4))-f31(k(1), k(2), k(3), k(4));...
            f21(k(1), k(2), k(3), k(4))-f32(k(1), k(2), k(3), k(4))];

% Combine all the constraints to one function handle 
constraints = @(k)deal(c(k),ceq(k)); 

% Only need the constraints to be satisfied, define a constant objective
% function
eq1 = @(k)0;

% A random starting guess, lower bound, upper bound 
% You can change this part to what you want
x0 = ones(1,4);
lb = [-inf, -inf, -inf, -inf];
ub = [inf, inf, inf, inf];

% No linear constraints 
A = [];
b = [];
Aeq = [];
beq = [];
[sol, fval, exitflag, output] = fmincon(eq1, x0, A, b, Aeq, beq, lb, ub, constraints);

解决方案

sol = [0.0116    0.5946   -0.3432    1.0064]

MATLAB 中的优化符号计算

Symbolic Computation with Optimisation in MATLAB

optimization

matlab

symbolic-math