Two-variable optimization with additional conditions

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.

[Bogdan Nikitchuk]

I assume, mathematically it should be some sort of the Lagrange multipliers method.

[John D'Errico]

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.

Answer 1

[Torsten]

Use "fmincon" and set the condition as a constraint in function "nonlcon".

Since you work with trigonometric functions, maybe restricting phi and theta by setting lower and upper bounds in the arrays "lb" and "ub" is an additional option.

[Bogdan Nikitchuk]

Thank you!

Answer 2

[Matt J]

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

[Bogdan Nikitchuk]

Thank you!