 Alternating proximal-gradient operator

Let $f:{\mathbb{R}}^p\times {\mathbb{R}}^q\to \mathbb{R}$ be a differentiable convex function whose full gradient is $L$-Lipschitz continuous, with $L>0$. The alternating update first solves the implicit $y$ step and then the explicit $x$ step:The implicit step is the proximal operator of $\tau f(x,\cdot )$, so it exists uniquely. The gradient bound must control the full joint variable; smoothness alone supplies no global constant $L$.

For two updates, let $(p,q)$ be the difference of their mixed points $(x,y^{+})$ and $(a,b)$ the corresponding full gradient difference. The input and output differences are $(p,q+\tau b)$ and $(p-\tau a,q)$. The mixed-point norm identity for alternating updates and Baillon–Haddad theorem giveHence $T$ is a $(\tau L/2)$-averaged operator for $0<\tau L<2$. Its fixed points have both block gradients zero and are exactly the global minimizers of $f$. If a minimizer exists, Browder convergence theorem for averaged operators gives norm convergence of iteration in the stated finite-dimensional setting.