A quick guide about solving equations (optimization problem)

Assume x is a vector of size n. It is a sample.

beta are some vectors of size n. Each beta_j is a vector. There are s different *beta*s.

a_j is a real number. a contains all s numbers related to a sample ( x ).

alpha is a known parameter. Lets assume it is one.

beta is known, so is x. By solving the following equation, we find a_hat which contains the proper coefficients. enter image description here I need to know whether this equation can be solved in MATLAB.

[Image Analyst]

How is this "a quick guide"? It doesn't seem to guide or help anybody. It doesn't look like a guide at all, but instead looks like your homework. Is it your homework? If so you should tag it as homework.

Answer 1

[John D'Errico]

A classic (and simple, even trivial) ridge regression problem, IF it were a 2 norm on a.

No toolbox would even be needed. Convert it into a simple regression problem, by augmenting your design matrix with sqrt(alpha)*eye(s). Solve using backslash. WTP?

As a 1 norm on a, so with mixed norms, you probably need to solve this using the optimization toolbox, so fminunc. Still easy enough. Read through the examples for fminunc. Still WTP?

Not sure why you feel the need to mix your norms anyway.

[Matin Kh]

Actually I solved it with LASSO, but your answer is quite close. Thank you.

Answer 2

[Matt J]

You can reformulate as a smooth problem in unknowns a(i), r(i)

 min norm(x-beta*a)^2 + alpha*sum(r)

with constraints

   -r(i)<=a(i)<=r(i)

This could be solved with quadprog or fmincon

Answer 3

[Marc]

Yes this problem can be solved with functions in the optimization toolbox.

[Matin Kh]

Are there any tutorials for this toolbox?