Convex optimization in Matlab

Question

I want to solve the following optimization problem:

Non-Latex: Given x and mu, find

argmin_p ||x-p||_2 s.t. ||p||_2 < mu.

Latex:

Given $\mathbf{x}$ and $\mu$, find

$\mathrm{argmin}_p \|\mathbf{x}-\mathbf{p}\|_2 \;\; \mathrm{s.t.}\;\;\|\mathbf{p}\|_2 \leq \mu$,

which is a convex function over a convex set. I have been using Matlab's fmincon but it is just too slow. Search engine results have so far brought me material that is much more theoretical than what I am looking for. I can't be the first person to want to solve this problem and was hoping to find an existing and efficient Matlab implementation.

Answer 1

cvx can handle this problem quite simply. Even if you're not familiar with it, the syntax is quite intuitive:

% test data
n = 1e4;
x = randn(n,1);
mu = 1;

cvx_begin
variable p(n)
minimize norm(x-p)
subject to
norm(p) <= mu
cvx_end

On my system this takes under a second for 10^4 variables.