how to solve a linear least square problem with non-linear constraint

Hi,

I am trying to solve a linear least square optimization problem but with a nonlinear constraint. Neither quadprog nor lsqlin could deal with nonlinear constraint. I could only try fmincon, but it is quite slow. Could anyone give me a suggestion?

The problem I am trying to solve is just like the standard form of quadratic programming problem, but with a constraint that x(1)^2+x(2)^2=1.

Answer 1

[John D'Errico]

It is a nonlinear constraint. What do you expect? Magic?

I'll offer a hint. Use a transformation. If you know that

x(1)^2 + x(2)^ = 1

then can you transform the problem, eliminating the constraint completely? (Yes.)

x(1) = cos(t)
x(2) = sin(t)

So optimize over the single parameter t. Your constraint is built directly into the problem, implicit in the transformation.

The problem will still be nonlinear, but you no longer have a constraint at all to deal with. Not magic, but far simpler to solve, and since you have one less parameter, a reduced space to search.

[Hengbo TANG]

Thank you for your reply.

Of course I could just eliminate the constraint by using one angle variable instead, but the objective problem would no longer be a quadratic form. I found that to solve a quadratic problem will be much more efficient than just define a nonlinear objective problem and use fmincon() to solve it.

Maybe there is an efficient approach consider my objective function?

[John D'Errico]

Of course a quadratic form will be efficient to solve. But you don't have a simple quadratic programming problem.

You might look at yalmip (on the FEX.)