Suppose there are several convex functions $f_i: \mathbb{R}^m \to \mathbb{R}, i = 1,2,\cdots,n$.
And suppose there are a series of convex optimization problems with $f_i$s as objective functions and with the same constraint functions.
Now we have a vector $\mathbf{w} = (w_1, w_2, \cdots, w_n)^T$ with $w_i \ge 0$ and $\sum_{i = 1}^n w_i = 1$.
A new convex optimization problem is formed with $\sum_{i=1}^n w_i f_i(x)$ as objective function and with the same constraint functions as before.
Can we conclude that, the solution $x^{\star}$ by solving this new convex optimization problem is the Pareto optimal of the new convex optimization problem?
Thank you!