Optimization involving dot and cross product of two unit length variables

Hi all,

I am trying to minimize the following function w.r.t. ,

, such that and

where are scalars and known. are known. the (.) and the(x) are the dot and cross products respectively.

Is there any way I could implement this using matlab?

Answer 1

[William Rose]

@Ahmad Hamad,

The function you want to minimze is below.

function y=AhmadsFunc(p)
%Ahmad Hamad's function to be minimized
s1=p(1:3);      %extract s1 from p
s2=p(4:6);      %extract s2 from p
j=1;            %insert desired values for j,k,ahat,D,B
k=2;
ahat=[1,2,3];
D=[4,5,6];
B=[7,8,9];
y=-j*dot(s1,s2)-dot(D,cross(s1,s2))-k*(dot(ahat,s1))^2-dot(B,s1)-dot(B,s2);
end

p=[s1;s2] is a 6x1 vector containing s1 and s2.

The nonlinear constraint function is

function [c,ceq]=AhmadsConstraints(p)
%Nonlinear constraints for Ahmad's problem.
%There are no inequality constraints.
%The equality constraints should each equal zero.
c=-1;                               %inequality constraint, must be <=0
ceq(1)=p(1)^2+p(2)^2+p(3)^2-1;      %||s1||=1
ceq(2)=p(4)^2+p(5)^2+p(6)^2-1;      %||s2||=1
end

The main program, below, makes an initial guess for p0=[s10;s20]. It calls fmincon(). Pass to fmincon() the initial guess, and the functon to be minimzed, and the function that returns the nonlinear contraints. fmincon() returns the best-fit parameter values.

%AhmadsMain.m   Ahmad Hamad & WCR
%Calls AhmadsFunc.m and AhmadsConstraints.m.
s10=[1;0;0]; s20=[0;1;0];
p0=[s10;s20];
p=fmincon(@AhmadsFunc,p0,[],[],[],[],[],[],@AhmadsConstraints);
fprintf('s1=%.3f,%.3f,%.3f, s2=%.3f,%.3f,%.3f\n',p);
fprintf('Length of s1=%.3f, length of s2=%.3f\n',norm(p(1:3)),norm(p(4:6)));

Here's the result I get.

>> AhmadsMain

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in

feasible directions, to within the default value of the optimality tolerance,

and constraints are satisfied to within the default value of the constraint tolerance.

<stopping criteria details>

s1=0.300,0.538,0.788, s2=0.521,0.484,0.703

Length of s1=1.000, length of s2=1.000

>>

You may get a different "best fit" if you change the initial guess, because fmincon() may find a local mimimum that is not a global minimum. Try different initial guesses, if this is a concern. I always do. I put the fitting routine inside a loop. I use a different initial guess on each pass. I save the fits from each pass and then, when the loop is done, I choose the best of them.

[Ahmad Hamad]

Thank you very much @William Rose

[William Rose]

You're welcome Ahmad.

Answer 2

[Matt J]

Yes, with fmincon()