= Implicit nonsmooth block in alternating proximal-gradient iteration
For $f(x,y)=h(x,y)+\phi(y)$, where $h$ is convex with an $L_h$-<Lipschitz gradient> and $\phi$ is convex, use a proximal $y$-step for the full section and an explicit $x$-step for $h$. Monotonicity of the two selected <subgradients> of $\phi$ combines with <cocoercivity> of $\nabla h$ to make the full update firmly nonexpansive for $0<\tau L_h\leq1$. Existence of a minimizer then gives convergence, even though the full objective is not smooth.
Back to article page
