A*x = b with maximal x

Hey guys, I am struggling with a rather easy mathematical problem:

I have a 3x2 Matrix, U = A\b gives me a solution with the minimal norm(U). But I want to have the maximum, where U(1,2,3) < 24000. Do you have an idea? Here is my code so far:

alpha = 12.34567;
dU = 1;

beta = [0 120 240];

A = [sind(beta(1)) sind(beta(2)) sind(beta(3)); cosd(beta(1)) cosd(beta(2)) cosd(beta(3))];

b = [tand(alpha) 1]';

% Rang = rank(A)
% 
U = A\b

Answer 1

[VK_11]

I found a solution, but it's not giving the right answers...

Startwert = [1,1,1];

[Uopt,UZeiger] = fmincon(@(U) Ufunktion(U),Startwert,[],[],[],[],[],[],@(U) Nebenbed(U,UZeiger,Umax,alpha,beta),optimset('Display','off','Algorithm','active-set'));
UZeiger = -UZeiger;

% Lösung mit Numerischer Methodik

function[UZeiger] = Ufunktion(U)
UZeiger = sum(U);
end
% Analog werden die Nebenbedingungen als Funktion definiert
function[c,ceq] = Nebenbed(U,UZeiger,Umax,alpha,beta)

% die Gleichungsnebenbedingungen
ceq = [sind(beta(1))*U(1) + sind(beta(2))*U(2) + sind(beta(3))*U(3) - tand(alpha);
       cosd(beta(1))*U(1) + cosd(beta(2))*U(2) + cosd(beta(3))*U(3) - 1;
      ];

% die Ungleichungsnebenbedingungen
c = UZeiger - 3*Umax; % U(2) - 24000;U(3) - 24000; % Maximalwerte der Spannung

end

[Torsten]

1. UZeiger and Umax are predefined values - so the inequality constraint is either satisfied or not. It does not influence the solution.

2. UZeiger= sum(U) is not adequate as objective function. Maybe you mean UZeiger = -sum(U.^2) ?

Best wishes

Torsten.

[VK_11]

The inequality constrait is actually being ignored. How can I fix this? Basically I want the maximal magnitude of the combined vector of Z, but the single vectors of U can't be over 24000. How is UZeiger predefined? I thought, that this is the variable I try to maximize.

This is my code:

alpha = 30;
Umax = 5;
% dU = 1;

beta = [0 120 240];

% XzuY = tand(alpha);

% A = [sind(beta(1)) sind(beta(2)) sind(beta(3)); cosd(beta(1)) cosd(beta(2)) cosd(beta(3))];
% 
% b = [tand(alpha) 1]';

% Rang = rank(A)
% 
% U = A\b

% Das Maximierungsproblem wird nun durch folgende Eingaben gelöst:
Startwert = [1,-1,1];

[Uopt,UZeiger] = fmincon(@(U) Ufunktion(U,beta),Startwert,[],[],[],[],[],[],@(U) Nebenbed(U,Umax,alpha,beta),optimset('Display','off','Algorithm','active-set'));
UZeiger = -UZeiger;

Uopt

% Lösung mit Numerischer Methodik
function[UZeiger] = Ufunktion(U,beta)

Winkel = beta/180*pi;

Z = U .* exp(1i*Winkel);
Y = Z(1) + Z(2) + Z(3);
UZeiger = -abs(Y);

end
% Analog werden die Nebenbedingungen als Funktion definiert
function[c,ceq] = Nebenbed(U,Umax,alpha,beta)

% die Gleichungsnebenbedingungen
ceq = [sind(beta(1))*U(1) + sind(beta(2))*U(2) + sind(beta(3))*U(3) - tand(alpha);
       cosd(beta(1))*U(1) + cosd(beta(2))*U(2) + cosd(beta(3))*U(3) - 1;
      ];

% die Ungleichungsnebenbedingungen
c = [abs(U(1)) - Umax; abs(U(2)) - Umax; abs(U(3)) - Umax;]; % Maximalwerte der Spannung

end

[Torsten]

Is this the problem you are trying to solve in a mathematical notation ?

max: x(1)^2 + x(2)^2 + x(3)^2

s.c.

x(1) <= 24000
x(2) <= 24000
x(3) <= 24000
sin(beta1)*x(1) + sin(beta2)*x(2) + sin(beta3)*x(3) = tan(alpha)
sin(beta1)*x(1) + sin(beta2)*x(2) + sin(beta3)*x(3) = 1

Best wishes

Torsten.

[VK_11]

Basically it is, though x(1,2,3) are 3 komplex vectors.

The vector that x(1,2,3) add up to must be maximized and can't be over 72000.

So I think you could describe the problem as this:

max: magnitude(Y)

Y = x(1) + x(2) + x(3) <= 72000

x(m) = X * exp(i*beta(m))

beta = [0 120 240]

alpha = angle(Y)

alpha = 30

X(1,2,3) (the length of x(1,2,3)) is the parameter that can be modified to get a solution for Y.

[Torsten]

What does Y <= 72000 mean for Y being a complex number ?

[VK_11]

Y is a complex number, it's magnitude must be smaller than 72000

[Torsten]

x(1)+x(2)+x(3) is not the magnitude of the vector x.

Sorry, but since I don't know the physical background of your problem, it seems that I'm not able to help you here.

Best wishes

Torsten.

[VK_11]

I solved my problem! As always it is the most stupid mistake after all. I didn't transform alpha into radians therefore only getting bad results. Thanks a lot for your help!

[Torsten]

I wonder where in your code you needed alpha in radians since you used "tand".

Anyhow ...

Best wishes

Torsten.

Answer 2

[Torsten]

Use "fmincon".

Best wishes

Torsten.

[VK_11]

Actually I already tried to use fmincon. Unfortunatly it can only be used for scalar values as far as I know. But I want three U to be maximized, or at least the length of the total vector.

[Torsten]

The length of the total vector is a scalar ...

Best wishes

Torsten.