Solving for scalar in matrix norm minimization

Is it possible in MATLAB to minimize argmin_alpha norm( X - alpha * Y , 1) (where X and Y are matrices)?

I want the following constraints:

alpha > 0, X - alpha * Y >= eps

Thanks

[Matt J]

The thing you propose to minimize X - alpha * Y _1 is not a scalar. Do you mean you want to minimize some squared norm of this difference? If so, which norm? L2? Frobenius?

[matlab user guy]

Sorry there was a problem with the text. This should be the matrix norm. The double bars were removed.

Answer 1

[Matt J]

If you have the Optimization Toolbox, you could also use fminimax, although that might be overkill for a simple scalar problem. Recall that the L1-norm of a matrix is its maximum absolute row sum.

[matlab user guy]

Thank you.

I have that toolbox. fminbnd doesn't seem to be working, but I'll check out fminimax.

Answer 2

[Matt J]

The system of linear inequalities

    X(i) - alpha * Y(i) >= eps

are equivalent to some 1D interval [alpha_lower, alpha_upper]. Once you find this interval, you can apply fminbnd.

The analysis needed to find the interval is simple, but you could let this FEX file do it for you,

http://www.mathworks.com/matlabcentral/fileexchange/30892-representing-polyhedral-convex-hulls-by-vertices-or--in-equalities