I am interested in the linear least square problem with the solution with the following constraints :
$$ \begin{alignat*}{3} \arg \min_{x} & \quad & \frac{1}{2} \left\| A x - b \right\|_{2}^{2} \\ \text{subject to} & \quad & \boldsymbol{1}^{T} x = 1 \\ & \quad & {x}_{i} \in \left[ 0, 1 \right], \; \forall i \end{alignat*} $$
Because of the second constraint, we know the optimal $x$ should lie in the convex region (simplex) whose vertices are the rows of the identity matrix $I_n$. I wanted to try an iterative algorithm where I start off with the $n$ points, namely the rows of $I_n$, compute the costs at each of these $n$ points, and then contract the convex region somehow, so as to reduce the volume, but still retain the optimal point(s). So I am looking for a set of rules that I can use to contract the convex region. For instance, I could compute the cost at the centroid of these $n$ points, and perhaps, replace the worst cost point with the centroid. That probably doesn't even guarantee convexity of the resulting region. So what are the set of rules that I can use that guarantee convexity, retains the optimal region, reduces the volume (possibly by a lot), and after a few iterations, results in a zero volume region (or a very very small region) that I can just pick the optimal point from?