Optimization when solving equation system

Good evening,

I'm trying to solve the following issue:

Given a vector (n=50 elements), which is expanded in known basis functions and known coefficients according to . I'd like to express v in an alternative known basis with unknown coefficients like . I obtain all coefficients via , in which the basis transformation matrix T follows from solution of the matrix equation with containing the as column vectors. This procedure works perfectly fine so far, with one exception: I'd like to ensure that , so the maximum value of the alternative expansion should never be smaller than the one of the original expansion.

Can this be done by optimization when solving for T in any way in Matlab?

Answer 1

[Matt J]

fmincon with a nonlinear constraint would probably work, although because your constraint is non-differentiable, strictly speaking the problem doesn't satisfy fmincon's assumptions. It might be worth doing a preliminary step, where you replace your max constraint with,

for some . Then, use that solution as the initial guess when you solve the original problem (which corresponds to).

[dj1du]

Thank you very much for your response! I just had a look at the function fmincon, but to be honest I'm not quite sure how to use it. Could you please give a more detailed explanation on its usage or even an example code, which applies to my specific problem?

[Matt J]

You would pose this as a constrained minimization problem:

The constraint is nonlinear, and so would be implemented by writing a nonlcon function,

https://www.mathworks.com/help/optim/ug/fmincon.html#busog7r-nonlcon