\begin{algorithm}{({\bf DR})}\label{alg:DR}
\begin{description}
\item[Step 1] ({\em Initialization}) Choose a parameter $\lambda\in\left]0,1\right[$ and the initial iterate $u^0$ arbitrarily. 
Choose a small parameter $\varepsilon>0$, and set $k=0$. 
\item[Step 2] ({\em Projection onto ${\cal B}$})  Set $u^- = \lambda u^{k}$. 
Compute $\widetilde{u} = P_{{\cal B}}(u^-)$ by using \eqref{eqn:projB}. 
\item[Step 3] ({\em Projection onto ${\cal A}$}) Set $u^- := 2\widetilde{u}-u^k$. 
Compute $\widehat{u} = P_{{\cal A}}(u^-)$ by using \eqref{eqn:projA} or Algorithm~\ref{alg:projA}.
\item[Step 4] ({\em Update}) Set $u^{k+1} := u^k + \widehat{u} - \widetilde{u}$.
\item[Step 5] ({\em Stopping criterion}) If $\|u^{k+1} - u^k\|_{L^\infty} \le \varepsilon$, then return $\widetilde{u}$ and stop.  
Otherwise, set $k := k+1$ and go to Step 2.
\end{description}
\end{algorithm}

\begin{algorithm}
{({\bf DR})}\begin{description}
\item[Step 1] ({\em Initialization}) Choose a parameter $\lambda\in\left]0,1\right[$ and the initial iterate $u^0$ arbitrarily.
Choose a small parameter $\varepsilon>0$, and set $k=0$.
\item[Step 2] ({\em Projection onto ${\cal B}$}) Set $u^- = \lambda u^{k}$.
Compute $\widetilde{u} = P_{{\cal B}}(u^-)$ by using \eqref{eqn:projB}.
\item[Step 3] ({\em Projection onto ${\cal A}$}) Set $u^- := 2\widetilde{u}-u^k$.
Compute $\widehat{u} = P_{{\cal A}}(u^-)$ by using \eqref{eqn:projA} or Algorithm~\ref{alg:projA}.
\item[Step 4] ({\em Update}) Set $u^{k+1} := u^k + \widehat{u} - \widetilde{u}$.
\item[Step 5] ({\em Stopping criterion}) If $\|u^{k+1} - u^k\|_{L^\infty} \le \varepsilon$, then return $\widetilde{u}$ and stop.
Otherwise, set $k := k+1$ and go to Step 2.
\end{description}
\end{algorithm}
{
    "title": "douglas--rachford algorithm for control-constrained minimum-energy   control problems",
    "id": "2210.17279",
    "abstract": "splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. in this paper we utilize the douglas--rachford (dr) algorithm to solve control-constrained minimum-energy optimal control problems. instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the dr algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. we derive general expressions for the projections and propose a numerical approach. we obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. we illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.",
    "categories": "math.oc",
    "doi": "",
    "created": "2022-10-31",
    "updated": "2024-01-11",
    "authors": [
        "regina s. burachik",
        "bethany i. caldwell",
        "c. yal\u00e7\u0131n kaya"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2210.17279"
}
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