Solving an optimization problem with several functions

I have the following 4 equations:

eq1=x11 - (9*2^(1/2)*u2)/(20*((2^(1/2)*u2)/2 - (5^(1/2)*v2)/5)) + (3*5^(1/2)*v2)/(50*((2^(1/2)*u2)/2 - (5^(1/2)*v2)/5))
                                                               eq2=x21 + (3*u2*v2)/(5*((2^(1/2)*u2)/2 - (5^(1/2)*v2)/5))
                                                    eq3=1/5 - (3*2^(1/2)*5^(1/2))/(50*((2^(1/2)*u2)/2 - (5^(1/2)*v2)/5))
 eq4=x22 - (3*2^(1/2)*u2)/(20*((2^(1/2)*u2)/2 - (5^(1/2)*v2)/5)) + (9*5^(1/2)*v2)/(50*((2^(1/2)*u2)/2 - (5^(1/2)*v2)/5))

for which I want to find the parameters

 x11,x21,x22,u2,v2

for which all the equations are 0 (or close to zero), subject to the constraints, and

So, I'm not sure how to approach this. I found the and the function, but they only accept a single function handle and I'm not sure if they are suited to solve this problem.

Is there a function in Matlab to solve such problems?

Any help appreciated!

Answer 1

[Stephan]

Hi,

one way to solve this is fsolve:

format long
x0 = [1 1 1 1 1];
options = optimoptions('fsolve', 'Algorithm', 'Levenberg-Marquardt');
result = fsolve(@equations, x0, options)
format short

function F = equations(x)
x11 = x(1);
x21 = x(2);
x22 = x(3);
u2 = x(4);
v2 = x(5);
F(1) = x11 - (9*2^(1/2).*u2)/(20*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5)) +...
    (3*5^(1/2).*v2)/(50*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
F(2) = x21 + (3.*u2.*v2)/(5*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
F(3) = 1/5 - (3*2^(1/2)*5^(1/2))/(50*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
F(4) = x22 - (3*2^(1/2).*u2)/(20*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5)) +...
    (9*5^(1/2).*v2)/(50*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
end

with result:

result =

   0.768206088209485   0.308533559026783   0.431793911790578   1.046940639379436  -0.465961843087484

Be careful! fsolve doesnt handle constraints. In this case all solutions except v2 are >0, so that your constraints are met.

EDIT:

You can use also use fmincon, by using a little trick:

format long
x0 = [1 1 1 1 1];
lb = [0 0 -inf -inf -inf];
ub = [inf inf inf inf inf];
% options = optimoptions('fmincon', 'Algorithm', 'Levenberg-Marquardt');
result = fmincon(@eqn1,x0,[],[],[],[],lb,ub,@nonlcon);
format short

function obj_fun = eqn1(x)
x11 = x(1);
% x21 = x(2);
% x22 = x(3);
u2 = x(4);
v2 = x(5);
obj_fun = x11 - (9*2^(1/2).*u2)/(20*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5)) +...
    (3*5^(1/2).*v2)/(50*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
end

function [c,ceq] = nonlcon(x)
x11 = x(1);
x21 = x(2);
x22 = x(3);
u2 = x(4);
v2 = x(5);
c = [];
ceq(1) = x11 - (9*2^(1/2).*u2)/(20*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5)) +...
    (3*5^(1/2).*v2)/(50*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
ceq(2) = x21 + (3.*u2.*v2)/(5*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
ceq(3) = 1/5 - (3*2^(1/2)*5^(1/2))/(50*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
ceq(4) = x22 - (3*2^(1/2).*u2)/(20*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5)) +...
    (9*5^(1/2).*v2)/(50*((2^(1/2).*u2)/2 - (5^(1/2).*v2)/5));
end

What happens here?

The first function takes your first equation as objective function to minimize. The second function defines the nonlinear constraints - here that all equations should be equal to zero. Since we use fmincon we can define lower and upper bounds to ensure that {x11, x22} > 0.

The result is:

result =

   0.733946078616734   0.360292286378543   0.466053921384332   0.970332937199620  -0.587089255763525

which meets the constraints. To check the results you could call the nonlcon-function with the results. Add this line after the fmincon call:

[~, check_result] = nonlcon(result) 

This should end near by zero.

check_result =

   1.0e-07 *

  -0.036086063998075   0.515513386978306  -0.002526874820497   0.036096726441226

This seems to be a good result.

Best regards

Stephan

[holistic]

Thank you Stephan! Still need to find a way to include those constraint though, since I have problems with much more equations and constraints

[Stephan]

See my edited answer. If it was useful for you, you can accept and/or like helpful answers. This is the easiest way to thak the volunteers here for their work.

[holistic]

Very helpful, thank you !!