\section{Simplifications when $n=1$}
\label{secnis1}
Suppose $n=1$, in which case we have either $1\in \Iback$ or $1\in\Iforw$. In
either case, using the convention discussed at the beginning of Section
\ref{subsecNote}, we have $w_1^k = 0$ for all $k$ and the affine function
defined in \eqref{hyper} becomes
\begin{align*}
\varphi_k(p) =\varphi_k(z) = \langle z-x^k,y^k\rangle,
\end{align*}
where we have written $x_1^k = x^k$ and $y_1^k = y^k$. 

If $1\in\Iback$, using $w_1^k = 0$, in the update on lines
\ref{lineaupdate}-\ref{lineBackwardUpdateY} of the algorithm yields
\begin{align*}
x^k + \rho^k y^k = z^k \qquad \text{ and } \qquad y^k\in T x^k,
\end{align*}
where we have written $T_1=T$ and $\rho_1^k = \rho^k$. 
This implies that
$
\varphi_k(z^k) =  \rho^k\|y^k\|^2.
$
Furthermore,
$
\nabla_z\varphi_k  = \gamma^{-1}y^k
$
and so $\|\nabla_z\varphi_k\|^2 = \gamma\cdot\gamma^{-2}\|y^k\|^2 = \gamma^{-1}\|y^k\|^2$. 
Therefore, using \eqref{eqUpds}, we have
\begin{align*}
z^{k+1} = z^k - \frac{\beta_k\varphi_k(p^k)}{\|\nabla\varphi_k\|^2}\nabla_z\varphi_k 
&=
z^k -\beta_k\rho^k y^k \\ &=
(1-\beta_k)z^k + \beta_k x^k =
(1-\beta_k)z^k
+
\beta_k\prox_{\rho^k}(z^k).
\end{align*}
Thus, when $n=1$ and $1\in\Iback$, projective splitting reduces to the relaxed
proximal point method of~\cite{EckBer92}; see also \cite[Theorem
23.41]{bauschke2011convex}. In fact, when one allows for approximate
evaluation of resolvents, projective splitting with $n=1\in\Iback$ reduces to
the hybrid projection proximal-point method of Solodov and Svaiter 
\cite[Algorithm 1.1]{solodov1999hybrid}. However, the error criterion
of~\cite[Eq.~(1.1)]{solodov1999hybrid} is more restrictive than the
conditions~\eqref{err1}--\eqref{err2} that we propose in
Assumption~\ref{assErr}.

On the other hand, if $1\in\Iforw$, considering lines
\ref{ForwardxUpdate}-\ref{ForwardyUpdate} or the algorithm with $w_1^k = 0$
yields
\begin{align*}
x^k = z^k - \rho^k T z^k \qquad \text{ and } \qquad y^k = T x^k.
\end{align*} 
Furthermore, we have 
\begin{align*}
\varphi_k(z^k) = \rho^k \langle T z^k,T x^k\rangle 
\qquad \text{ and } \qquad 
\nabla_z \varphi_k = \gamma^{-1}T x^k.
\end{align*}
Therefore,
\begin{align*}
z^{k+1} 
&= 
z^k 
-
 \frac{\beta_k\varphi_k(p^k)}{\|\nabla\varphi_k\|^2}\nabla_z\varphi_k 
 =
 z^k - 
 \frac{\beta_k\rho^k \langle T z^k,T x^k\rangle }
 { \|Tx^k\|^2}
 Tx^k.
\end{align*}
Thus, the method reduces to
\begin{align*}
x^k &= z^k - \rho^kTz^k
\\
z^{k+1} &= z^k - \tilde{\rho}^k Tx^k,
\end{align*}
where
\begin{align}\label{eqStepAd}
\tilde{\rho}^k= \frac{\beta_k\rho^k \langle T z^k,T x^k\rangle}{\|Tx^k\|^2}.
\end{align}
This is the unconstrained version of the extragradient method
\cite{korpelevich1977extragradient}. When $\beta_k=1$, the stepsize
\eqref{eqStepAd} corresponds to the extragradient stepsize proposed by Iusem
and Svaiter in \cite{iusem1997variant}.  Furthermore, the backtracking
linesearch for the extragradient method also proposed in
\cite{iusem1997variant} is almost equivalent in the unconstrained case to the
linesearch we proposed for processing individual operators within projective
splitting in \cite{johnstone2018projective}, except that Iusem and Svaiter use
a more restrictive termination condition (perhaps necessary because they also
account for the constrained case of the extragradient method).

While these observations may be of interest in their own right, they also have
implications for the convergence rate analysis. In particular, that projective
splitting reduces to the proximal point method suggests that the $\bigO(1/k)$
ergodic convergence rate for \eqref{ProbOpt} derived in Section \ref{secFunc}
cannot be improved beyond a constant factor. This is because the same rate is
unimprovable for the proximal point method under the assumption that the
stepsize is bounded from above and below \cite{guler1991convergence}.
{"config": "arxiv", "file": "1806.03920/nIs1.tex"}