Hello every one, I have a question related to approximation of functions based on norms

How can I approximate this function and its constraint using L1, L2, and L∞ to perform the optimization process using fmincon MATLAB function?

Objective function F*w+f=min

Subject to F*w+f ≥b

F is a matrix (7x35)

f is a vector (7x1)

b is the vector (7x1)

w is the vector of parameters needed for optimization (35x1)

Answer 1

[Matt J]

You would not use fmincon. For L1, you could use minl1lin

https://www.mathworks.com/matlabcentral/fileexchange/52795-constrained-minimum-l1-norm-solutions-of-linear-equations

For L2, you would use lsqlin,

https://www.mathworks.com/help/optim/ug/lsqlin.html

For L-infinity, you would use fminimax,

https://www.mathworks.com/help/optim/ug/fminimax.html

[Ahmed Galal]

Thanks for your answer, but what about approximation. I mean that I need first approximate the function based on different norms then apply in yours optimization functions. So what are the function approximations based on L1, L2, and L∞

[Ahmed Galal]

I am sorry, I understand now. Thanks

[Ahmed Galal]

What should I do if some constraints included in the norms like the attached example. Please find the attached example

The first constraint is L∞-NORM

The second constraint is L1-NORM

[Matt J]

That's a much different question. You should post it in a separate thread.

Answer 2

[Bruno Luong]

l1 and linfinity can be formulate as linear programming, so LINPROG is better.

[Matt J]

Yes, and in the L1 case, formulating as a linear program is done for you by minl1lin.

[Bruno Luong]

It can also deal with extra constraints from Ahmed's addendum.

Anyway, some people seems to be more interested in getting the fish rather than learn how to fish.

[Matt J]

The first constraint in Ahmed's addendum is non-convex. It looks difficult to recast that as a linear program.

[Ahmed Galal]

Thank you very much sir Bruno Luong for helping, also for your kind words.

Thanks also extending for sir Matt.