= Alternating proximal-gradient operator
{title2=$y-y^+=\tau\nabla_yf(x,y^+),\quad x-x^+=\tau\nabla_xf(x,y^+)$}
For a jointly smooth <convex function> $f(x,y)$, update $y$ implicitly by its proximal minimization at fixed $x$, then update $x$ explicitly with the gradient at the mixed point $(x,y^+)$. The combined map is $(\tau L/2)$-averaged for $0<\tau L<2$, as follows by applying <cocoercivity> at two mixed points. Its fixed points are precisely the minimizers. If a minimizer exists, averaged iteration proves convergence.
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