I would like to use the convex combination of two convex functions, but I am not able to formulate the problem because of the no multiplication rule in CVX.

Is there any way to do this?

Here is the problem: I generate a sample set of two Gaussians, and trying to estimate the means and at which time-instance it should belong to which mean. For this I am trying to use the convex combination of two norms. Here is the code:

```
T = 200;
K=2;
MU1 = [5];
SIGMA1 = [1.5];
MU2 = [0];
SIGMA2 = [.5];
r1 = mvnrnd(MU1,SIGMA1,T/K);
r2 = mvnrnd(MU2,SIGMA2,T/K);
r = [r1; r2];
cvx_begin
variable m1(size(MU1));
variable m2(size(MU2));
variable a1(T,1);
variable a2(T,1);
minimize(sum(norms(r-m1,2,2)'*a1 + norms(r-m2,2,2)'*a2))
subject to
a1 >= zeros(size(a1));
a2 >= zeros(size(a2));
a1 + a2 == ones(size(a1));
cvx_end
```

As a result I get

??? Error using ==> cvx.mtimes at 153

Disciplined convex programming error:

Cannot perform the operation {convex}*{affine}

Which is correct given that in general {convex}*{affine} function is not convex, but given the conditions that both a1 and a2 is non-negative and they sum up to one, the objective function should be a convex combination of two convex functions, which (I think) is convex.

Am I mistaken somewhere?