optimization of a function with array of parameters using matlab solver ga

Question

fima v 2022-11-11

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1848618-optimization-of-a-function-with-array-of-parameters-using-matlab-solver-ga

编辑： Torsten 2022-11-12

在 MATLAB Online 中打开

Hello, i i am trying to optimize function S, so it will get values between 0 and 10 using its E_opt parameter

coefficients which shown bellow in the expression of S.

I defined optimvar for each sub cell of the E_opt, and used solve as shown bellow.

but then i get an error on optimvar saying E_opt(1) is not a valid matlab name.

how do we run such optimization so i will get optimized array of parameters E_opt?

Thanks.

clc
clear all
f=6;
lambda=300/6;
theta_in_rad=(0/180)*pi;
theta_out_rad=(70/180)*pi;
x=linspace(-15*lambda,15*lambda,1000);
y=linspace(-15*lambda,15*lambda,1000);
[xx,yy]=meshgrid(x,y);
k=2*pi/lambda;
E_i=1;
E_r=E_i*sqrt(cos(theta_in_rad)/cos(theta_out_rad))
eta=377;
D_x=1.064*lambda;
k_x=k*sin(theta_in_rad)+2*pi/D_x;
k_y=sqrt(k^2-k_x.^2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
n=[2,3,4,5,6,7,8];
k_x_n=k*sin(theta_in_rad)+(2*pi.*n)/D_x;
k_y_n=sqrt(k^2-k_x_n.^2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E_opt(1)=optimvar('E_opt(1)','Lower',0,'Upper',+1);
E_opt(2)=optimvar('E_opt(2)','Lower',0,'Upper',+1);
E_opt(3)=optimvar('E_opt(3)','Lower',0,'Upper',+1);
E_opt(4)=optimvar('E_opt(4)','Lower',0,'Upper',+1);
E_opt(5)=optimvar('E_opt(5)','Lower',0,'Upper',+1);
E_opt(6)=optimvar('E_opt(6)','Lower',0,'Upper',+1);
E_opt(7)=optimvar('E_opt(7)','Lower',0,'Upper',+1);
S=@(E_opt)abs(E_i.*exp(-j*k*xx*sin(theta_in_rad)+j*k*yy*cos(theta_in_rad))+E_r.*exp(-j*k_x*xx-j*k_y*yy)+E_opt(1).*exp(-j*k_x_n(1)*xx-j*k_y_n(1)*yy)+E_opt(2).*exp(-j*k_x_n(2)*xx-j*k_y_n(2)*yy)+E_opt(3).*exp(-j*k_x_n(3)*xx-j*k_y_n(3)*yy)+E_opt(4).*exp(-j*k_x_n(4)*xx-j*k_y_n(4)*yy)+E_opt(5).*exp(-j*k_x_n(5)*xx-j*k_y_n(5)*yy)+E_opt(6).*exp(-j*k_x_n(6)*xx-j*k_y_n(6)*yy)+E_opt(7).*exp(-j*k_x_n(7)*xx-j*k_y_n(7)*yy).*conj((j*k*cos(theta_in_rad))*E_i.*exp(-j*k*xx*sin(theta_in_rad)+j*k*yy*cos(theta_in_rad))+(-j*k_y)*E_r.*exp(-j*k_x*xx-j*k_y*yy)+(-j*k_y_n(1))*E_opt(1).*exp(-j*k_x_n(1)*xx-j*k_y_n(1)*yy)+(-j*k_y_n(2))*E_opt(2).*exp(-j*k_x_n(1)*xx-j*k_y_n(2)*yy)+(-j*k_y_n(3))*E_opt(3).*exp(-j*k_x_n(3)*xx-j*k_y_n(3)*yy)+(-j*k_y_n(4))*E_opt(4).*exp(-j*k_x_n(4)*xx-j*k_y_n(4)*yy)+(-j*k_y_n(5))*E_opt(5).*exp(-j*k_x_n(5)*xx-j*k_y_n(5)*yy)+(-j*k_y_n(6))*E_opt(6).*exp(-j*k_x_n(6)*xx-j*k_y_n(6)*yy)+(-j*k_y_n(7))*E_opt(7).*exp(-j*k_x_n(7)*xx-j*k_y_n(7)*yy)));
S_max=@(E_opt)max(abs(E_i.*exp(-j*k*xx*sin(theta_in_rad)+j*k*yy*cos(theta_in_rad))+E_r.*exp(-j*k_x*xx-j*k_y*yy)+E_opt(1).*exp(-j*k_x_n(1)*xx-j*k_y_n(1)*yy)+E_opt(2).*exp(-j*k_x_n(2)*xx-j*k_y_n(2)*yy)+E_opt(3).*exp(-j*k_x_n(3)*xx-j*k_y_n(3)*yy)+E_opt(4).*exp(-j*k_x_n(4)*xx-j*k_y_n(4)*yy)+E_opt(5).*exp(-j*k_x_n(5)*xx-j*k_y_n(5)*yy)+E_opt(6).*exp(-j*k_x_n(6)*xx-j*k_y_n(6)*yy)+E_opt(7).*exp(-j*k_x_n(7)*xx-j*k_y_n(7)*yy).*conj((j*k*cos(theta_in_rad))*E_i.*exp(-j*k*xx*sin(theta_in_rad)+j*k*yy*cos(theta_in_rad))+(-j*k_y)*E_r.*exp(-j*k_x*xx-j*k_y*yy)+(-j*k_y_n(1))*E_opt(1).*exp(-j*k_x_n(1)*xx-j*k_y_n(1)*yy)+(-j*k_y_n(2))*E_opt(2).*exp(-j*k_x_n(1)*xx-j*k_y_n(2)*yy)+(-j*k_y_n(3))*E_opt(3).*exp(-j*k_x_n(3)*xx-j*k_y_n(3)*yy)+(-j*k_y_n(4))*E_opt(4).*exp(-j*k_x_n(4)*xx-j*k_y_n(4)*yy)+(-j*k_y_n(5))*E_opt(5).*exp(-j*k_x_n(5)*xx-j*k_y_n(5)*yy)+(-j*k_y_n(6))*E_opt(6).*exp(-j*k_x_n(6)*xx-j*k_y_n(6)*yy)+(-j*k_y_n(7))*E_opt(7).*exp(-j*k_x_n(7)*xx-j*k_y_n(7)*yy))));
S_min=@(E_opt)min(abs(E_i.*exp(-j*k*xx*sin(theta_in_rad)+j*k*yy*cos(theta_in_rad))+E_r.*exp(-j*k_x*xx-j*k_y*yy)+E_opt(1).*exp(-j*k_x_n(1)*xx-j*k_y_n(1)*yy)+E_opt(2).*exp(-j*k_x_n(2)*xx-j*k_y_n(2)*yy)+E_opt(3).*exp(-j*k_x_n(3)*xx-j*k_y_n(3)*yy)+E_opt(4).*exp(-j*k_x_n(4)*xx-j*k_y_n(4)*yy)+E_opt(5).*exp(-j*k_x_n(5)*xx-j*k_y_n(5)*yy)+E_opt(6).*exp(-j*k_x_n(6)*xx-j*k_y_n(6)*yy)+E_opt(7).*exp(-j*k_x_n(7)*xx-j*k_y_n(7)*yy).*conj((j*k*cos(theta_in_rad))*E_i.*exp(-j*k*xx*sin(theta_in_rad)+j*k*yy*cos(theta_in_rad))+(-j*k_y)*E_r.*exp(-j*k_x*xx-j*k_y*yy)+(-j*k_y_n(1))*E_opt(1).*exp(-j*k_x_n(1)*xx-j*k_y_n(1)*yy)+(-j*k_y_n(2))*E_opt(2).*exp(-j*k_x_n(1)*xx-j*k_y_n(2)*yy)+(-j*k_y_n(3))*E_opt(3).*exp(-j*k_x_n(3)*xx-j*k_y_n(3)*yy)+(-j*k_y_n(4))*E_opt(4).*exp(-j*k_x_n(4)*xx-j*k_y_n(4)*yy)+(-j*k_y_n(5))*E_opt(5).*exp(-j*k_x_n(5)*xx-j*k_y_n(5)*yy)+(-j*k_y_n(6))*E_opt(6).*exp(-j*k_x_n(6)*xx-j*k_y_n(6)*yy)+(-j*k_y_n(7))*E_opt(7).*exp(-j*k_x_n(7)*xx-j*k_y_n(7)*yy))));
cam=fcn2optimexpr( S,E_opt);
camMax=fcn2optimexpr( S_max,E_opt);
camMin=fcn2optimexpr( S_min,E_opt);
prob = optimproblem("Objective",camMax);
prob.Constraints.camUpper=camMax<=10;
prob.Constraints.camLower=camMin>=0;
[sol,fval] = solve(prob,"Solver","ga");

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fima v 2022-11-12

Hello ,my E_opt array of variables needed to be complex numbers

is there a way to change the command bellow so each cell will be complex so the amplitude will be between 1 to 7 and the phase is between 0 to 220 degrees?

Thanks.

E_opt = optimvar('E_opt',7,'Lower',0,'Upper',1);

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Answer 1

Torsten 2022-11-11

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1848618-optimization-of-a-function-with-array-of-parameters-using-matlab-solver-ga#answer_1097153

在 MATLAB Online 中打开

E_opt = optimvar('E_opt',7,'Lower',0,'Upper',+1);

instead of

E_opt(1)=optimvar('E_opt(1)','Lower',0,'Upper',+1);
E_opt(2)=optimvar('E_opt(2)','Lower',0,'Upper',+1);
E_opt(3)=optimvar('E_opt(3)','Lower',0,'Upper',+1);
E_opt(4)=optimvar('E_opt(4)','Lower',0,'Upper',+1);
E_opt(5)=optimvar('E_opt(5)','Lower',0,'Upper',+1);
E_opt(6)=optimvar('E_opt(6)','Lower',0,'Upper',+1);
E_opt(7)=optimvar('E_opt(7)','Lower',0,'Upper',+1);

and

[sol,fval] = solve(prob);

instead of

[sol,fval] = solve(prob,"Solver","ga");

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Walter Roberson 2022-11-11

You wish to "optimize" a function but have constaints on what the value of the function is.

If you are minimizing a function S and wish to constrain the minimized value of the function is at least 0, then there are two possibilities:

The function is naturally non-negative and the >= 0 constraint has no effect. In this case, minimizing the function without the constraint will return a positive minima for the function; OR
The function crosses 0, and the >= 0 constraint does have an effect. In this case you wish to know the location where it reaches 0. But you can get that by solving the function for 0.

This leads to the following strategy if you are doing minimization

minimize the function without the constraints
if the value at the minima is greater than 10 then there is no solution that is consistent with your constraints. Solving for the function equalling 10 will not work because if the minima were 10 or lower it would have been found by the minimizer
If the value of the unconstrained minima is between 0 and 10 (inclusive) then that fits within your constraints, so accept the location without further processing
If the value of the unconstrained minima is less than 0, then attempt to solve for the equations equalling 0 exactly. If the equations always return negative than this will fail. If the equations cross 0 then solving for 0 will find such a location, and that will satisfy the constraints that 0 <= S <= 10 because it will be a point at which S == 0 by design.

You can use a very similar strategy if you are maximizing instead of minimizing, except that in that case you would potentially be solving for S == 10 rather than S == 0

fima v 2022-11-12

Hello ,my E_opt array of variables needed to be complex numbers

is there a way to change the command bellow so each cell will be complex so the amplitude will be between 1 to 7 and the phase is between 0 to 220 degrees?

Thanks.

E_opt = optimvar('E_opt',7,'Lower',0,'Upper',1);

Torsten 2022-11-12

编辑：Torsten 2022-11-12

Read the warning in the MATLAB documentation:

https://de.mathworks.com/help/optim/ug/complex-numbers-in-optimization-toolbox-solvers.html

Warning

The problem-based approach does not support complex values in an objective function, nonlinear equalities, or nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result might be incorrect.

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