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

Abstract

For a symmetric positive semidefinite linear system of equations $$\mathcal{Q}{{\varvec{x}}}= {{\varvec{b}}}$$ , where $${{\varvec{x}}}= (x_1,\ldots ,x_s)$$ is partitioned into s blocks, with $$s \ge 2$$ , we show that each cycle of the classical block symmetric Gauss–Seidel (sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form $$\frac{1}{2}\Vert {{\varvec{x}}}-{{\varvec{x}}}^k\Vert _\mathcal{T}^2$$ , where $$\mathcal{T}$$ is a symmetric positive semidefinite matrix related to the sGS decomposition of $$\mathcal{Q}$$ and $${{\varvec{x}}}^k$$ 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 convex composite QP (CCQP) with an additional possibly nonsmooth term in $$x_1$$ , i.e., $$\min \{ p(x_1) + \frac{1}{2}\langle {{\varvec{x}}},\,\mathcal{Q}{{\varvec{x}}}\rangle -\langle {{\varvec{b}}},\,{{\varvec{x}}}\rangle \}$$ , where $$p(\cdot )$$ is a proper closed convex function. Based on the block sGS decomposition theorem, we extend the classical block sGS method to solve 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 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.