Two-variable optimization with additional conditions
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Let me describe the setup. Let us say I have two unit vectors (all of the components are known real numbers), and with , and . I also have some two-variable function . I would like to sole the following optimization problem
with the additional condition i.e.
The optimization itself is not a problem, however, I have no idea how to not only solve the optimization problem but also satisfy the last condition.
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John D'Errico
2024년 2월 2일
It usually would be, except you should never be writing your own optimization code for your work, when it is already available. Especially if you are not an expert on the subject. Use FMINCON, as suggested. Or use other tools that allow nonlinear constraints. GA, for example.
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Matt J
2024년 2월 2일
편집: Matt J
2024년 2월 2일
The constraint is equivalent to minimizing over the intersection of the unit sphere and a plane through the origin (normal to n0). This is a great circle of the unit sphere. You should therefore just parametrize the great circle in terms of some angle t and then rewrite your function f() as a 1 dimensionsal function of t. This reduces the problem to a 1-variable problem with no constraints, which you can solve with fminbnd.
basis=null(n0(:).'); %basis for plane of great circle
fobj=@(t) objective(t,n0); %objective re-written as function of t
t_optimal = fminbnd( fobj , 0, 2*pi);
[~,phi,theta] = fobj(t_optimal);
function [fval,phi,theta] = objective(t,basis)
n1=basis*[sin(t);cos(t)];
[phi,theta]=cart2sph(n1(1), n1(2), n1(3));
fval=f(theta,phi);
end
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