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Take $F=(F^{\ast }{)}^{\ast }$ with the usual proper closed convex assumptions, so the Fenchel-Moreau theorem identifies its conjugate with the supplied $F^{\ast }$. Put $z^k=u^k-\tau D^T{v}^k$. The first update is $w^{k+1}={\mathrm{prox}}_{F^{\ast }/\tau }(z^k/\tau )$. By Moreau decomposition, the second update isAlso $\tau (D^T{v}^k+w^{k+1})=u^k-u^{k+1}$, so the argument in the final update contains $u^{k+1}-\tau (D^T{v}^k+w^{k+1})=2u^{k+1}-u^k$. Eliminating the auxiliary variable givesThis is the primal-dual hybrid gradient method with extrapolation parameter one and the primal update performed first. Its saddle point function is $F(u)+⟨Du,v⟩-G(v)$, corresponding to the primal objective $F(u)+G^{\ast }(Du)$.

For proper lower semicontinuous convex functions and a nonempty saddle-point set in these finite-dimensional spaces, the standard convergence of primal-dual hybrid gradient theorem gives the sufficient parameter conditionHere $\Vert D{\Vert }_2$ is the operator norm, or largest singular value. For $D\ne 0$, one possible choice is $\tau =\sigma =\eta /\Vert D{\Vert }_2$ with $0<\eta <1$; for $D=0$ any positive steps satisfy the condition. The existence assumption is necessary: step sizes alone cannot guarantee convergence to a saddle point that does not exist.