I need to solve
\begin{alignat}{1} & \min_{x}\|Ax - b\|^2_{2}, \\ \mathrm{s.t.} & \quad\sum_{i}x_{i} = 1, \\ & \quad x_{i} \geq 0, \quad \forall{i}. \end{alignat}
I think it is a quadratic problem which should be solvable with CVXOPT, but I can't work out how.
I wrote a full answer (below the line) before discovering CVXPY, which (like CVX for MATLAB) does all the hard stuff for you and has a very short example almost identical to yours here. You only need to replace the relevant line with
p = program(minimize(norm2(A*x-b)),[equals(sum(x),1),geq(x,0)])
My old answer, doing it the harder way with CVXOPT:
Following Geoff's suggestion to square your objective function gives
$$\|Ax-b\|^2_2 = \langle x^TA^T - b^T,Ax-b\rangle = x^TA^TAx - b^TAx - x^TAb - b^Tb$$
Of course, all terms are scalars, so you can transpose the third one and drop the last one (as it doesn't depend on $x$ and therefore won't change which $x$ gives you a minimum, though you will need to add it back in after solving in order to get the correct value of your objective) to obtain $$ x^TA^TAx - b^T(A + A^T)x$$ This (including your constraints) has the form of a quadratic program, as given in the CVXOPT documentation here, where there is also example code for solving such a problem.
Instead of the problem you solved, solve
\begin{alignat}{1} & \min_{x}\|Ax - b\|^{2}_{2}, \\ \mathrm{s.t.} & \quad\sum_{i}x_{i} = 1, \\ & \quad x_{i} \geq 0, \quad \forall{i}. \end{alignat}
This problem is a differentiable, convex, nonlinear optimization problem that can be solved in CVXOPT, IPOPT, or any other convex optimization solver.
