A block symmetric Gauss-Seidel decomposition theorem for convex composite quadratic programming and its applications

For a symmetric positive semidefinite linear system of equations $\mathcal{Q} {\bf x} = {\bf b}$, where ${\bf x} = (x_1,\ldots,x_s)$ is partitioned into $s$ blocks, with $s \geq 2$, we show that each cycle of the classical block symmetric Gauss-Seidel (block sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form $\frac{1}{2} \| {\bf x}-{\bf x}^k \|_{\mathcal T}^2$, where ${\mathcal T}$ is a symmetric positive semidefinite matrix related to the sGS decomposition and ${\bf x}^k$ is the previous iterate. By leveraging on such a connection to optimization, we are able to extend the result (which we name as the block sGS decomposition theorem) for solving a convex composite QP (CCQP) with an additional possibly nonsmooth term in $x_1$, i.e., $\min\{ p(x_1) + \frac{1}{2}\langle {\bf x},\, \mathcal{Q} {\bf x} \rangle -\langle {\bf b},\, {\bf x}\rangle\}$, where $p(\cdot)$ is a proper closed convex function. Based on the block sGS decomposition theorem, we are able to extend the classical block sGS method to solve a CCQP. In addition, our extended block sGS method has the flexibility of allowing for inexact computation in each step of the block sGS cycle. At the same time, we can also accelerate the inexact block sGS method to achieve an iteration complexity of $O(1/k^2)$ after performing $k$ block sGS cycles. As a {fundamental} building block, the block sGS decomposition theorem has played a key role in various recently developed algorithms such as the inexact semiproximal {ALM/ADMM} for linearly constrained multi-block convex composite conic programming (CCCP), and the accelerated block coordinate descent method for multi-block CCCP.