I have a quadratic program that looks like this:
For fixed vector $b$, and matrices $A_1, ..., A_n$, Find column vectors $x_1, ..., x_n$ that minimize $\sum_{i=1}^n x_i ^T 1 1^T x_i$ subject to $\sum_{i=1}^n A_i x_i = b$ and $x_i\in \mathcal{X_i}$, where $\mathcal{X}_i$ is a convex set.
I've been able to solve the problem using two ways. The first is using proximal gradient methods, and the second is using ADMM (both first order methods), but both have a very slow convergence and I have to choose a "correct" learning rate.
I was wondering if there is a way to use second order methods on this problem or a practical method for speeding up the convergence.