Let $\alpha=(\alpha\_1,\ldots,\alpha\_m)\subset\mathbb R^m\_+$ and $\beta=(\beta\_1,\ldots,\beta\_n)\subset\mathbb R^n\_+$ be given and satisfy


$$\sum\_{i=1}^m \alpha\_i =1 = \sum\_{j=1}^n\beta\_j.$$


Define $\Pi(\alpha,\beta)\subset \mathbb R^{mn}\_+$ be the subset consisting $c=(c\_{ij})\_{1\le i\le m, 1\le j\le n}$ satisfying for all $i=1,\ldots, m$ and $j=1,\ldots, n$


$$\sum\_{j=1}^n c\_{ij} =\alpha\_i \quad \mbox{ and } \quad \sum\_{i=1}^m c\_{ij} =\beta\_j.$$


Consider the minimisation problem as follows:


$$\min\_{c\in \Pi(\alpha,\beta)}~~ \sum\_{j=1}^n\left[\sum\_{i=1}^m c\_{ij}|x\_i|^2 -\frac{\big|\sum\_{i=1}^m c\_{ij}x\_i\big|^2}{\beta\_j}\right],$$


where $x\_1,\ldots, x\_m\in \mathbb R^d$ are fixed parameters, and $|\cdot|$ denotes the standard Euclidean norm. Is there a "well developed" numerical scheme for the above optimisation problem?


PS : "well developed" means


1. The numerical scheme must converge (as it is known that the above minimisation problem admits a minimiser);
2. There are checkable conditions such that the numerical scheme may converge to a minimiser (or at least a local minimiser);
3. An analysis on the convergence rate is furnished.



A variant of (discrete) optimal transport problem
The minimum here can be found exactly, in a finite number of steps.


Indeed, the target function is concave. So, its minimum on the (compact) (transportation) polytope $P:=\Pi(\alpha,\beta)$ is attained at one of the extreme points of $P$. For a construction and a characterization of all extreme points of $P$, see [this answer by Brendan McKay](https://mathoverflow.net/a/75905/36721).


