How is the solution set of the optimization problem
$$\min \frac{1}{2} x^T A x - x^T b \text{ s.t. } x \in K,$$
where $A$ is symmetric positive semi-definite and $K$ not convex related to the set
$$\{x\ |\ \forall i : x_i = \operatorname{arg min}_{x \in K} \frac{1}{2} x_i^T A_{ii} x_i - x_i^T (b_i - \sum_{j \neq i} A_{ij} x_j)\},$$
where $A_{ii}$ are diagonal blocks of $A$? The diagonal blocks $A_{ii}$ are known to be positive definite.
