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%   mte08-IKY.tex  Tue., 07.12.11  <===  MTE08.tex  <===  sample_mte08.tex 
%                                  <===  Golden-dual.tex 2006.10.6
%    IKY := Iwamoto+Kira+Yasuda
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%    The final version is completed on Thursday, Jan. 10, 2008
%    This final version's pfd-file is sent 
%                   to Prof. Matsuhisa on Thursday, Jan. 10, 2008
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\documentclass[leqno]{article}
%\usepackage{mte08e}
\usepackage{mte08e2}  % IWAMOTO var.
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage[mathscr]{eucal}
%%%%%%%%%%%%%%%%      Definition Of Macro     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\Sup{\displaystyle\mathop{\rm Sup}}
\def\Inf{\displaystyle\mathop{\rm Inf}}
\def\inf{\displaystyle\mathop{\rm inf}}
\def\Max{\displaystyle\mathop{\rm Max}}
\def\max{\displaystyle\mathop{\rm max}}
\def\max{\displaystyle\mathop{\rm max}}
\def\min{\displaystyle\mathop{\rm min}}
\def\dsum{\displaystyle\sum}
\def\dlim{\displaystyle\lim}
\def\dfrac{\displaystyle\frac}
%
\def\mdot{\hspace{-0.48mm}\cdot\hspace{-0.61mm}}
\def\ndot{\hspace{-0.35mm}\cdot\hspace{-0.48mm}}
\def\eql{\hspace{0.46mm}&=&\hspace{0.46mm}}
%\def\eql{\hspace{-1.46mm}&=&\hspace{-1.46mm}}
\def\dql{\hspace{0.49mm}=\hspace{0.49mm}}
\def\cql{\hspace{0.89mm}=\hspace{0.89mm}}
\def\nge{\hspace{-1.46mm}&\ge&\hspace{-1.46mm}}
\def\mge{\hspace{0.66mm}\ge\hspace{0.66mm}}
\def\nle{\hspace{-1.46mm}&\le&\hspace{-1.46mm}}
\def\mle{\hspace{0.66mm}\le\hspace{0.66mm}}
%  
\def\proof{{\it \noindent Proof.} \hspace{2ex}}
\def\boxend{$\blacksquare$}
%\def\boxend{ \hspace*{\fill} $\blacksquare$	 \\ }
%\def\boxend{ \hspace*{\fill} $\Box$	 \\ }
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% 
%
%% The numbering of theorems, lemmas, etc.
%% may be set by the author
\newdefin{dfn}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newdefin{remark}{Remark}
\newdefin{example}{Example}
\newdefin{lem}{Lemma}   %  by Iwamoto

\begin{document}
\keywords{Golden ratio, dual, dynamic optimization, dynamic programming, Lagrange multiplier}
\mathclass{Primary 90C39; Secondary 49L20.}

\abbrevauthors{S. Iwamoto, A. Kira and M. Yasuda}
\abbrevtitle{Golden Duality}

\title{Golden Duality in Dynamic Optimization}
%
%  Golden-dual.tex 2006.10.6 ==> Fibo-dual.tex 2006.8.7
%  Golden Duality in Dynamic Optimization
%  The Golden dual in dynamic optimization
%  or
%   The Golden solution in primal and dual dynamic optimization \\
%  Golden-dual.tex  ==> Fibo-dual.tex 2006.8.7
%
\author{Seiichi Iwamoto,\,~Akifumi Kira}
\address{Department of Economic Engineering, Graduate School of Economics, Kyushu University\\
Hakozaki 6-19-1, Higashi-ku, Fukuoka 812-8581, Japan\\
E-mail: iwamoto@en.kyushu-u.ac.jp, ec207004@s.kyushu-u.ac.jp}
\thanks{This research is supported by the Grants in Aid for Scientific 
Research (B) No.~17340030 in the Japan Society for the Promotion of Sciences.}

\author{Masami Yasuda}
\address{Department of Mathematics and Informatics, Faculty of Science, 
Chiba University\\
Yayoi-cho 1-33, Inage-ku, Chiba 263-8522, Japan\\
E-mail: yasuda@math.s.chiba-u.ac.jp}


\maketitleMTE


\begin{abstract}
This paper discusses the dual of infinite-variable quadratic minimization (primal) problems from a view point of Golden ratio. 
We consider two pairs of primal and dual (maximization) problems. 
One pair yields the Golden duality. 
While the minimum value function is Golden quadratic and the minimum point constitutes a Golden path, so is the maximum value function and the maximum point does such another. 
The other yields the inverse-Golden duality. 
While the minimum value function is inverse-Golden quadratic and the minimum point constitutes the same Golden path, so is the maximum value function and the maximum point does the same such another. 
\vskip 5mm
\end{abstract}

%
%\section{Introduction.} 
%
%This sample paper for the Proceedings of the Second KOSEN Workshop MTE 2008 is intended to facilitate the publication process. The publisher strongly encourages submission of papers prepared using \TeX. {\bf After acceptance of the paper,} you will be asked to send both the {\bf final} {\TeX} source file and its PDF file (i.e., do not send {\tt .dvi} files) by e-mail to the editor: 
%
%\medskip 
%\noindent
%{\bf Prof. Takashi Matsuhisa}\\
% e-mail: {\bf confmte@ge.ibaraki-ct.ac.jp} and {\bf WorkshopMTE@aol.com} \\
%Department of Natural Sciences, Ibaraki National College of Technology\\
%Nakane 866, Hitachinaka-shi, Ibaraki 312-8508, Japan.
%\medskip
%
%\noindent
%{\bf Paper submission deadline: January 10, 2008}\\
%\medskip
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%It is essential that the file exactly match the submitted hard copy. If you make any (even minor) changes of text, send also  the {\bf revised} {\TeX} source file with its PDF file to the editorial committee. All versions of {\LaTeX} are accepted. {\LaTeX} style files are available. The publisher will create page proofs for final review only by the editor not by the authors. The authors should carefully check their draft papers when they send them to the editor in the proceedings.
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%\longdiv{1234}{5}\longdiv{333.33}{2.915}
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%[]�݁�������݁́ȁɁʁˁ́ځہ܁ށ߁���������������������
%�A�A�B�C�D�E�F�G�H�I�J�K�L�M�N�P_�S�T�U�V�W�X�Y�Z�[�\-/�_�`�a�b�d�c�e�f�g�h�i�j%�k�l�m�n~�a|�c�k�l�m�p�p�o�q�r�s�t�u�v�w�x�y�z�{�}�|�~
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%****************************************************************************
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%\part{Nondeterministic Processes}
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%*******************************************************************
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\section{Introduction}
%
%******************************************************************
~

The Golden ratio is one of the most beautiful numbers. The desire for optimality is inherent for humans. One minimization leads to the other maximization, which arrives at a duality. We direct our attention to both the Golden ratio and the duality. A duality of fine features is shown.

In this paper, we are concerned with dynamic optimization problems of infinitely many  variables from a viewpoint of Golden duality. 
We take two typical quadratic minimization (primal) problems with initial condition and associate each problem with a quadratic maximization (dual) problem with transversality condition. The two pairs of primal and dual problems have interesting characteristics. 
%
%  The Golden ratio has a lot of beautiful characteristics.   
%                                  By Collin throuh Kira Mon Jan 28 2008
%
As for the first pair, the minimum value function is Golden quadratic and the minimum point constitutes a Golden path, while so is the maximum value function and the maximum point does such another.
As for the second, the minimum value function is inverse-Golden quadratic and the minimum point constitutes the same Golden path, while so is the maximum value function and the maximum point does the same such another. 
%*******************************************************************
%
\section{Duality}
%
%******************************************************************
~

A real number 
$$ \phi = \dfrac{1 + \sqrt{5}}{2} \, \thickapprox \, 1.618 $$
is called {\it Golden number} \cite{abbp,dunl,wals}. It is the larger of the two solutions to quadratic equation (QE)  
\begin{equation}  
    x^{2} - x - 1 = 0.    \label{eqn:fi1}  
\end{equation}
Sometimes QE (\ref{eqn:fi1}) is called {\it Fibonacci} or {\it Golden}. 
The Golden QE has two real solutions: $ \phi $ and its {\it conjugate}  $ \overline{\phi} := 1 - \phi $. We note that 
$$    \phi + \overline{\phi} \dql 1,~~~~~\phi \ndot \overline{\phi} \dql -1. $$
Further we have 
\begin{eqnarray*}
  & \phi^{-1} \dql \phi - 1, ~~~~~(\overline{\phi})^{-1} \dql -\phi &  \\
  &\phi^{-1} + (\overline{\phi})^{-1} \dql -1,~~~~~\phi^{-1} \ndot (\overline{\phi})^{-1} \dql -1 &  \\
  & \phi^{2} \dql 1+ \phi, ~~~~~\overline{\phi}^{2} \dql 2 - \phi&  \\
  &\phi^{2} + \overline{\phi}^{2} \dql 3,~~~~~\phi^{2} \ndot \overline{\phi}^{2} \dql 1. &
\end{eqnarray*}


A quadratic function $ v(x) = ax^{2} $ is called {\it Golden} (resp. {\it inverse-Golden}) if $ a = \phi $ (resp. $ \phi^{-1}$). In this section, we consider two pairs of primal and dual problems. One pair yields a duality for the Golden quadratic function.  The other pair yields a duality for the inverse-Golden quadratic function. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Golden duality}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
~

We take an interval $[0, x] $, where $x > 0.$  
Let us consider the set of all divisions of the interval $[0, x] $. Each division is specified by an inner point $ y \in [0, x] $, which splits the interval $[0, x] $ into two intervals $[0, y] $ and $[y, x] $. 
A point $ (2 - \phi)x $ splits the interval into two intervals $[0, (2 - \phi)x] $ and $[(2 - \phi)x, x] $. A point $ (\phi -1)x $ splits it into $[0, (\phi -1)x] $ and $[(\phi -1)x, x] $. In either case, the length constitutes the Golden ratio $ (2 - \phi) : (\phi -1) = 1 : \phi. $ Thus both divisions are the {\it Golden section} \cite{abbp,dunl,wals}. 

\begin{dfn} \cite{e87} A sequence $ x: \{0, 1, \ldots \} \to R^{1} $ is called {\it Golden path} if and only if either
\begin{equation}   
\dfrac{\,x_{t+1}\,}{\,x_{t} \,} \dql \phi - 1~~~~~\mbox{or}~~~~~\dfrac{\,x_{t+1}\,}{\,x_{t} \,} \dql 2 - \phi.  \nonumber %  \label{eqn:dg1}
\end{equation}
\end{dfn}

\begin{lem} \cite{e87} A Golden path $ x$ is either 
\begin{equation}   
 x_{t} \dql x_{0}(\phi - 1)^{t}~~~~~\mbox{or}~~~~~x_{t} \dql x_{0}(2 - \phi)^{t}.  \nonumber % \label{eqn:dg3}
\end{equation}
\end{lem}

%
%%%%%%%%%%%%%%%%%%%%%%    �����o�H�̃O���t   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  �ɒl���@Extremize == Locally maximize and/or locally minimize
%%%%%%%%%%%%%%%%%%%%%%  �����o�H�̃O���t�i�����j gol-traj-1.tex  %%%%%%
\begin{center}
\input{gl-traje-dis46.tex}
%%%%%%%%%%%%%%%%%%%%%%
%\begin{center}
Fig.~1~~Golden paths $ x = c(\phi - 1)^{t}~~~c = 1, 2, 3 $ 
%\end{center}
%%%%%%%%%%%%%%%%%%%%%%  �����o�H�̃O���t gol-traj-1-c.tex  %%%%%%
\input{gl-traje-dis64.tex}
%%%%%%%%%%%%%%%%%%%%%%
%\begin{center}
Fig.~2~~Golden paths  $ x = c(2 - \phi)^{t}~~~c = 1, 2, 3 $ 
\end{center}
%   Adjusted by A.Kira  2008.1.15
We remark that 
\begin{equation}
(\phi - 1)^{t} = \phi^{-t},~~~~~~(2 - \phi)^{t} = (1 + \phi)^{-t} 
 \nonumber 
\end{equation} 
where
$$ \phi - 1 \dql \phi^{-1} \,\thickapprox \, 0.618,~~~~~2 - \phi  \dql (1 + \phi)^{-1} \,\thickapprox \, 0.382  $$ 
%
\vskip 3mm

Let $ R^{\infty} $ be the set of all sequences of real values : 
$$ R^{\infty} = \{ x = (x_{0}, x_{1}, \ldots , x_{n}, \ldots ) \, | \,x_{n} \in R^{1}~~n = 0, 1, \ldots \, \}.  $$
%
We consider a primal problem$^1$\vfootnote{$^1$}{As for corresponding finite variable problems see \cite{iwae}, and as for their dual and others see \cite{iw71, iw72, iwrr, iwae, iwtw}.} on $ R^{\infty} $ :  
%
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=0}^{\infty}\left[x_{n}^{2} + (x_{n} - x_{n+1})^{2} \right] \\ 
({\rm P}_{1})~~~& &{\rm subject~~ to}\hspace{3ex} {\rm (i)}~~ x \in R^{\infty} \\
& &\hspace*{20.5mm} {\rm (ii)}~~x_{0} = c
\end{eqnarray*}
where $ c \in R^{1} $.
%
A dual problem is a maximization problem of $ \lambda = (\lambda_{0}, ~\lambda_{1}, ~\ldots , ~\lambda_{n}, \ldots) \in R^{\infty} $:  
%
\begin{eqnarray*}
& &{\rm Maximize} \hspace{3mm} c^{2} + c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right]  \\ 
({\rm D}_{1})~~~& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ \lambda \in R^{\infty}  \\[1.05mm]
& &\hspace*{18.5mm}  {\rm (ii)}~~~ \dlim_{n \to \infty}\lambda_{n} = 0.
\end{eqnarray*}
%
We note that both problems contain a common series $ \dsum_{n=0}^{\infty}\left[y_{n}^{2} + (y_{n} - y_{n+1})^{2} \right]$. In either problem, we are concerned with the finite convergence case : $ \dsum_{n=0}^{\infty}\left[y_{n}^{2} + (y_{n} - y_{n+1})^{2} \right] < \infty$. This implies that $ \dlim_{n \to \infty}y_{n} = 0. $ 
In Section 3, we will see that the additional transversality condition {\rm (ii)} enables us to make dual of $({\rm P}_{1})$ without difficulty. Therefore, the transversality condition may be removed from the constraints. 
\vskip 3mm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{theorem} \label{thm:a3}  {\rm (Golden duality)}~ \rm{(i)}~~ The primal problem 
$ ({\rm P}_{1}) $ has the minimum value 
$ m  = \phi c^{2} $ at the point  
\begin{equation}
 \hat{x} = c\left( 1,~ (2 - \phi),~ \ldots ,~ (2 - \phi)^{n} , \ldots \right).  \nonumber 
\end{equation}
%
 \rm{(ii)}~~ The dual problem 
$({\rm D}_{1}) $ has the maximum value 
 $ M   = \phi c^{2} $ at the point 
$$ \lambda^{*} \dql 2 \phi^{-1} c \left( 1,~ (2 - \phi),~ \ldots ,~ (2 - \phi)^{n} , \ldots \right). $$
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem
\vskip 3mm

We make an observation about the two optimal solutions. Both the minimum point and the maximum point constitute a {\it Golden path} and both the minimum value function and the maximum value function are the identical {\it Golden quadratic value function}. 
What a beautiful duality this is! Thus, the duality is called {\it Golden duality}. The proof will be given throughout the discussion in Section 3. 

We see that the minimum solution of $ ({\rm P}_{1}) $ yields the maximum solution of $ ({\rm D}_{1}) $. Let $ ({\rm P}_{1}) $ have the minimum value $ m  = \phi c^{2} $ at the minimum point $ \hat{x}. $ Then we have 

\begin{eqnarray*}
& &\Max\left[ c^{2} + c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right] \right] \\[2.56mm]
\eql \Max_{\lambda_{0}} \left[ c^{2} + c \lambda_{0} - \frac{1}{\,4\,} \min_{\{\lambda_{n}\}_{n \ge 1}} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right] \right]   \\[2.56mm]
\eql \Max_{\lambda_{0}} \left[ c^{2} + c \lambda_{0} - \frac{1}{\,4\,}\phi \lambda_{0}^{2} \right] \\[2.56mm]
\eql \phi c^{2} ~~~~~\mbox{for}~~~~\lambda_{0} \dql 2 \phi^{-1} c
\end{eqnarray*}
where the minimum is attained at 
$$ \left( \hat{\lambda}_{1},~ \hat{\lambda}_{2},~ \ldots ,~ \hat{\lambda}_{n} , \ldots \right)
 = \lambda_{0} \left( 2 - \phi,~ (2 - \phi)^{2},~ \ldots ,~ (2 - \phi)^{n} , \ldots \right). $$ 
%
\vskip 3mm 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{theorem} \label{thm:gd1} The functional equation 
\begin{eqnarray*}
f(c) \eql \Max_{\lambda \in R^{1}} \left[ c^{2} + c \lambda - \frac{1}{\,4\,}f(\lambda) \right]~~~~c \in R^{1} 
\end{eqnarray*}
has the Golden quadratic maximum value function $ f(c) \dql \phi c^{2} $ for maximum point function  $ \hat{\lambda}(c) \dql 2\phi^{-1} c. $ This is a unique continuous function. 
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem
\proof 
It is easily verified that $f$ with $ \hat{\lambda} $ satisfies the functional equation. The uniqueness is shown through the well known method in dynamic programming \cite[Ch.4]{bell}, because of $ \left|-\dfrac{1}{4}\right| < 1$ (see also \cite{walt}).  \boxend
\vskip 3mm 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{corollary} \label{thm:ge1} The functional equation 
\begin{eqnarray*}
f(c) \eql \Max_{\lambda \in R^{1}} \left[ c \lambda + \frac{1}{\,4\,}\lambda^{2} - \frac{1}{\,4\,}f(\lambda) \right]~~~~c \in R^{1} 
\end{eqnarray*}
has the Golden quadratic maximum value function $ f(c) \dql \phi c^{2} $ for maximum point function  $ \breve{\lambda}(c) \dql 2\phi c. $ This is a unique continuous function. 
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem

\vskip 3mm 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{corollary} \label{cor:ge1} The functional equation 
\begin{eqnarray*}
\frac{1}{\,4\,}f(c) \eql \Max_{\lambda \in R^{1}} \left[\lambda^{2} + c \lambda - f(\lambda) \right]~~~~c \in R^{1} 
\end{eqnarray*}
has the Golden quadratic maximum value function $ f(c) \dql \phi c^{2} $ for maximum point function  $ \tilde{\lambda}(c) \dql \dfrac{\,\phi \,}{\, 2\,} c. $ 
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Inverse-Golden duality}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
~

Second we consider a primal problem 
%
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=0}^{\infty}\left[(x_{n} - x_{n+1})^{2} + x_{n+1}^{2} \right] \\ 
({\rm P}_{2})~~~& &{\rm subject~~ to}\hspace{3ex} {\rm (i)}~~ x \in R^{\infty} \\
& &\hspace*{20.5mm} {\rm (ii)}~~x_{0} = c
\end{eqnarray*}
and a dual problem 
%
\begin{eqnarray*}
& &{\rm Maximize} \hspace{3mm} c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right]  \\ 
({\rm D}_{2})~~~& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ \lambda \in R^{\infty}  \\[1.05mm]
& &\hspace*{18.5mm}  {\rm (ii)}~~~ \dlim_{n \to \infty}\lambda_{n} = 0
\end{eqnarray*}
%
where $ c \in R^{1}. $ We note that a difference between $ ({\rm P}_{1}) $ and  $ ({\rm P}_{2}) $ is constant :
$$ \dsum_{n=0}^{\infty}\left[x_{n}^{2} + (x_{n} - x_{n+1})^{2} \right] \dql x_{0}^{2} + \dsum_{n=0}^{\infty}\left[(x_{n} - x_{n+1})^{2} + x_{n+1}^{2} \right]. 
$$
The difference $ x_{0}^{2} = c^{2} $ is also preserved between $ ({\rm D}_{1}) $ and  $ ({\rm D}_{2}) $. This enables us to obtain a duality in terms of inverse-Golden number $ \phi^{-1} = \phi - 1 $ as follows.
\vskip 3mm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{theorem} \label{thm:as3} {\rm (Inverse-Golden duality)}~ \rm{(i)}~~ The primal problem 
$ ({\rm P}_{2}) $ has the minimum value 
$ m  = \phi^{-1} c^{2} $ at the point  
\begin{equation}
 \hat{x} = c\left( 1,~ (2 - \phi),~ \ldots ,~ (2 - \phi)^{n} , \ldots \right).  \nonumber 
\end{equation}
%
 \rm{(ii)}~~ The dual problem 
$({\rm D}_{2}) $ has the maximum value 
 $ M = \phi^{-1} c^{2} $ at the point 
%  A correction 2008.1.15 by S. Iwamoto   2 \phi c ==>  2 \phi^{-1} c 
$$ \lambda^{*} \dql 2 \phi^{-1} c \left( 1,~ (2 - \phi),~ \ldots ,~ (2 - \phi)^{n} , \ldots \right). $$
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem
\vskip 3mm

We remark that both optimal points constitute a {\it Golden path} and that both optimal value functions are the same --- the inverse-Golden quadratic value function ---. This is a Golden duality, too. The Golden paths $ \hat{x} $ and $ \lambda^{*} $ are the same ones in $ ({\rm P}_{1}) $ and $ ({\rm D}_{1}) $, respectively.

Further the minimum solution of $ ({\rm P}_{2}) $ yields the maximum solution of $ ({\rm D}_{2}) $. Let $ ({\rm P}_{2}) $ have the minimum value $ m  = \phi^{-1} c^{2} $ at the minimum point $ \hat{x}. $ Then we have 

\begin{eqnarray*}
& &\Max\left[ c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right] \right] \\[2.56mm]
\eql \Max_{\lambda_{0}} \left[ c \lambda_{0} - \frac{1}{\,4\,} \lambda_{0}^{2} - \frac{1}{\,4\,} \min_{\{\lambda_{n}\}_{n \ge 1}} \dsum_{n=0}^{\infty} \left[ \left (\lambda_{n} - \lambda_{n+1} \right)^{2} + \lambda_{n+1}^{2} \right]  \right]   \\[2.56mm]
\eql \Max_{\lambda_{0}} \left[ c \lambda_{0} - \frac{1}{\,4\,}\lambda_{0}^{2} - \frac{1}{\,4\,}\phi^{-1} \lambda_{0}^{2} \right] \\[2.56mm]
\eql \phi^{-1} c^{2} ~~~~~\mbox{for}~~~~\lambda_{0} \dql 2 \phi^{-1} c
\end{eqnarray*}
where the minimum is attained at 
$$ \left( \hat{\lambda}_{1},~ \hat{\lambda}_{2},~ \ldots ,~ \hat{\lambda}_{n}, \ldots \right)
 = \lambda_{0} \left( 2 - \phi,~ (2 - \phi)^{2},~ \ldots ,~ (2 - \phi)^{n} , \ldots \right). $$ 
\vskip 3mm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{theorem} \label{thm:gf3} The functional equation 
\begin{eqnarray*}
g(c) \eql \Max_{\lambda \in R^{1}} \left[ c \lambda - \frac{1}{\,4\,} \lambda^{2} - \frac{1}{\,4\,}g(\lambda) \right]~~~~c \in R^{1} 
\end{eqnarray*}
has the inverse-Golden quadratic maximum value function $ g(c) \dql \phi^{-1} c^{2} $ for maximum point function  $ \breve{\lambda}(c) \dql 2\phi^{-1} c.$ This is a unique continuous function. 
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem
\vskip 3mm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{corollary} \label{thm:gf1} The functional equation 
\begin{eqnarray*}
g(c) \eql \Max_{\lambda \in R^{1}} \left[ -c^{2} + c \lambda - \frac{1}{\,4\,}g(\lambda) \right]~~~~c \in R^{1} 
\end{eqnarray*}
has the inverse-Golden quadratic maximum value function $ g(c) \dql \phi^{-1} c^{2} $ for maximum point function  $ \hat{\lambda}(c) \dql 2\phi c. $ This is a unique continuous function. 
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem
\vskip 3mm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{corollary} \label{thm:gf5} The functional equation 
\begin{eqnarray*}
\frac{1}{\,4\,}g(c) \eql \Max_{\lambda \in R^{1}} \left[ -c \lambda - \lambda^{2} - g(\lambda) \right]~~~~c \in R^{1} 
\end{eqnarray*}
has the inverse-Golden quadratic maximum value function $ g(c) \dql \phi^{-1} c^{2} $ for maximum point function  $ \tilde{\lambda}(c) \dql \dfrac{\,\phi \,}{\, 2\,} c. $ 
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Lagrangean Method}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
~

In this section we show how the Lagrangean method derives a maximization (dual) problem from the minimization (primal) problem.

Let us reconsider the primal problem 
%
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=0}^{\infty}\left[x_{n}^{2} + (x_{n} - x_{n+1})^{2} \right] \\ 
({\rm P}_{1})~~~& &{\rm subject~~ to}\hspace{3ex} {\rm (i)}~~ x \in R^{\infty} \\
& &\hspace*{20.5mm} {\rm (ii)}~~x_{0} = c. 
\end{eqnarray*}
%
We introduce a sequence of variables $ u = \{u_{0}, ~u_{1}, ~\ldots ,~ u_{n}, ~\ldots\}$ by 
 $$ u_{k} = x_{k+1} - x_{k}.  $$
Then $ ({\rm P}_{1}) $ is formulated into a quadratic minimization under a linear constraint 
%
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=0}^{\infty} \left (x_{n}^{2} + u_{n}^{2} \right) \\ 
({\rm P}^{\prime}_{1})~~~& &{\rm subject~~ to}\hspace{3ex} {\rm (i)}~~ x_{n+1} = x_{n} + u_{n} ~~~ n \ge 0 \\
& &\hspace*{20.5mm} {\rm (ii)}~~x_{0} = c.
\end{eqnarray*}

Let us now solve this problem through a Lagrangean multiplier's method. We introduce a sequence of variables $ \lambda = \{\lambda_{0}, ~\lambda_{1}, ~\ldots ,~ \lambda_{n}, ~\ldots\}$ with the property $ \dlim_{n \to \infty}\lambda_{n} = 0 $, which is called a {\it Lagrange multiplier}. 
Let us construct the Lagrangean 
\begin{eqnarray}
L(x, u , \lambda) \eql \dsum_{n=0}^{\infty} \left [x_{n}^{2} + u_{n}^{2} - \lambda_{n}(x_{n+1} - x_{n} - u_{n}) \right].        \nonumber 
\end{eqnarray}
Then it has the partial derivatives 
\begin{eqnarray}
L_{x_{n}} \eql 2x_{n} + \lambda_{n} - \lambda_{n-1}~~~~~~n \ge 1   \nonumber   \\[2.01mm]
L_{u_{n}} \eql 2u_{n} + \lambda_{n}~~~~~~~n \ge 0   \nonumber    \\[2.01mm]
L_{\lambda_{n}} \eql x_{n+1} - x_{n} - u_{n}~~~~~~~~n \ge 0.   \nonumber 
\end{eqnarray}
Here we notice that the Lagrangean is not one for a {\it regular} (a finite $n$ variables with a finite $m$ constraints) extremal problem. The problem has a countably infinte variables under a countably infinite linear constraints. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Begin of Lemma
\begin{lem} \label{lem:ft1}
Let $ (x, u) $ be an extremum point. Then there exists a $ \lambda $ satisfying the condition that all the partial derivatives at point $ (x, u, \lambda) $ vanish: 
\begin{eqnarray}
L_{x_{n}} \eql 0~~~~ n \ge 1,~~~~~L_{u_{n}} \dql L_{\lambda_{n}} \dql 0~~~~ n \ge 0.  \label{eqn:lm1} 
\end{eqnarray}
\end{lem} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Lemma
\proof  
Let $ \hat{x} = (\hat{x}_{n})_{n \ge 0}, \hat{u} = (\hat{u}_{n})_{n \ge 0} $ be an extremum point for $ ({\rm P}^{\prime}_{1})$. We take any large positive integer $ N $ and consider a finite-truncated conditional minimization problem of $ x = (x_{n})_{0}^{N}, u = (u_{n})_{0}^{N} $ :
%
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=0}^{N} \left (x_{n}^{2} + u_{n}^{2} \right) \\ 
({\rm T}_{N})~~~& &{\rm subject~~ to}\hspace{3ex} {\rm (i)}~~ x_{n+1} = x_{n} + u_{n} ~~~ 0 \le n \le N-1 \\
& &\hspace*{20.5mm} {\rm (ii)}~~\hat{x}_{N+1} = x_{N} + u_{N} \\
& &\hspace*{20.0mm} {\rm (iii)}~~x_{0} = c.
\end{eqnarray*}
This has $ (2N + 1)$ variables $x_{1}, \ldots , x_{N},\, u_{0}, \ldots , u_{N} $ and $(N + 1)$ linear constraints. 
Let us construct the Lagrangean $ L^{N} $ by
\begin{eqnarray}
L^{N}(x, u, \lambda) \eql \dsum_{n=0}^{N-1} \left [x_{n}^{2} + u_{n}^{2} - \lambda_{n}(x_{n+1} - x_{n} - u_{n}) \right]         \nonumber \\[1.56mm]
& & \, + ~ x_{N}^{2} + u_{N}^{2} - \lambda_{N}(\hat{x}_{N+1} - x_{N} - u_{N})~~~\mbox{for}~~\lambda = (\lambda_{n})_{0}^{N}.        \nonumber
\end{eqnarray}
Then the point $ (\hat{x}_{n})_{1}^{N}, (\hat{u}_{n})_{0}^{N} $ is also an extremum point for the truncated problem. It satisfies the linear independent constraint qualification \cite{kutu}. Therefore, Lagrange Multiplier Theorem (for a regular problem) implies that there exists a $ (\lambda^{*}_{n})_{0}^{N} $ such that $ (\hat{x}_{n})_{1}^{N}, (\hat{u}_{n})_{0}^{N};(\lambda^{*}_{n})_{0}^{N}  $ satisfies 
\begin{eqnarray}
L^{N}_{x_{n}} \eql 0~~~~ 1 \le n \le N,~~~~~L^{N}_{u_{n}} \dql L^{N}_{\lambda_{n}} \dql 0~~~~ 0 \le n \le N.  \nonumber 
\end{eqnarray}
Thus we have 
\begin{eqnarray}
2\hat{x}_{n} + \lambda^{*}_{n} - \lambda^{*}_{n-1} \eql  0~~~~~~1 \le n \le N   \nonumber   \\[1.56mm]
2\hat{x}_{n} + \lambda^{*}_{n} \eql  0~~~~~~0 \le n \le N  \nonumber    \\[1.56mm]
\hat{x}_{n+1} - \hat{x}_{n} - \hat{u}_{n} \eql  0~~~~~~0 \le n \le N.   \nonumber 
\end{eqnarray}
Since $ N $ is arbitrarily large, we conclude that there exists a $ (\lambda^{*}_{n})_{n \ge 0} $ such that $ (\hat{x}_{n})_{n \ge 1}, (\hat{u}_{n})_{n \ge 0}$\\$;(\lambda^{*}_{n})_{n \ge 0} $ satisfies (\ref{eqn:lm1}). 
This completes the proof. \boxend  

%This is a system of $(3n+3)$ equations for $(3n+3)$ unknowns $ (x, u , \lambda) $. 

Then (\ref{eqn:lm1}) is equivalent to 
\begin{eqnarray}
2x_{n} \eql -(\lambda_{n} - \lambda_{n-1})~~~~n \ge 1  \nonumber    \\[1.01mm]
2u_{n} \eql -\lambda_{n}~~~~~n \ge 0   \nonumber    \\[1.01mm]
 u_{n} \eql x_{n+1} - x_{n}~~~~~n \ge 0. \nonumber 
\end{eqnarray}
Now we solve this equivalent system in the following. 
Deleting $ u $ and $ \lambda $, we get a system of linear equations:
\begin{eqnarray}
x_{0} \eql c   \nonumber \\[-2.36mm]
& &  \label{eqn:hf3}   \\[-2.36mm]
x_{n-1} - 3x_{n} + x_{n+1} \eql 0 ~~~~n \ge 1.   \nonumber 
\end{eqnarray}
Thus we have 
\begin{eqnarray}
u_{n} \eql x_{n+1} - x_{n} \nonumber  \\[-2.36mm]  
& &     \label{lag:hs1}   \\[-2.36mm]
\lambda_{k} \eql -2(x_{n+1} - x_{n}).   \nonumber 
\end{eqnarray}

We see that Eq.(\ref{eqn:hf3}) has the solution$^2$\vfootnote{$^2$}{A general solution of (\ref{eqn:hf3}) is $ x_{n} = A (2 - \phi)^{n} + B (1 + \phi)^{n}$, where $ A + B = c.$ The case $ A = c, B = 0 $ attains a minimum value for $({\rm P}_{1})$.}
\begin{equation}
  x_{n} \dql c (2 - \phi)^{n} ~~~~n \ge 0.   \label{sol:hs3}
\end{equation}
where $ \phi $ is the Golden number (ratio). 
Thus we have 
\begin{eqnarray}
u_{n} \eql -\dfrac{\,c\,}{\,\phi\,} (2 - \phi)^{n} \nonumber  \\[-1.36mm]  
& &     \label{lag:hs5}   \\[-1.36mm]
\lambda_{n} \eql \dfrac{\,2c\,}{\,\phi\,} (2 - \phi)^{n}.   \nonumber 
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Lemma
\begin{lem} 
The solution $ (x,\, u) $ in (\ref{sol:hs3}),(\ref{lag:hs5}) is a minimum point for $({\rm P}^{\prime}_{1}) $. Hence, $ x $ is a minimum point for $({\rm P}_{1}) $.  
\end{lem} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Lemma
\proof  
We show that the $ (x,\, u) $ is a minimum point. 
Let $ (X,\, U) $ be any solution satisfying 
\begin{eqnarray*}
& & X_{n+1} = X_{n} + U_{n} ~~~~~ n \ge 0 \\
& & X_{0} = c.
\end{eqnarray*}
Here we take the $ \lambda $ in (\ref{lag:hs5}). First we see that for any $n$-stage process
\begin{eqnarray}
\dsum_{k=0}^{n} \left (x_{k}^{2} + u_{k}^{2}\right) 
\eql \dsum_{k=0}^{n} \left [x_{k}^{2} + u_{k}^{2} - \lambda_{k}(x_{k+1} - x_{k} - u_{k}) \right]   \nonumber \\
\eql c^{2} + c\lambda_{0} - \lambda_{0}x_{1} +  \dsum_{k=1}^{n-1} \left [x_{k}^{2} - \lambda_{n}(x_{k+1} - x_{k})\right] + x_{n}^{2} + \lambda_{n}x_{n}  \nonumber \\
& & ~+ \dsum_{k=0}^{n} \left(u_{k}^{2} + \lambda_{k} u_{k}\right) - \lambda_{n}x_{n+1}  \nonumber \\
%
\eql c^{2} + c\lambda_{0} +  \dsum_{k=1}^{n} \left [x_{k}^{2} + (\lambda_{k} - \lambda_{k-1}) x_{k} \right]    + \dsum_{k=0}^{n} \left(u_{k}^{2} + \lambda_{k} u_{k}\right) - \lambda_{n}x_{n+1} \nonumber \\
%
\eql c^{2} + c\lambda_{0}  - \frac{1}{4} \dsum_{k=1}^{n} \left (\lambda_{k} - \lambda_{k-1} \right)^{2} - \frac{1}{4}  \dsum_{k=0}^{n} \lambda_{k}^{2} - \lambda_{n}x_{n+1} \nonumber \\
& &~ + \dsum_{k=1}^{n} \left [x_{k} + \frac{1}{2}(\lambda_{k} - \lambda_{k-1}) \right]^{2} + \dsum_{k=0}^{n} \left(u_{k} + \frac{1}{2} \lambda_{k}\right)^{2}.  \label{eqn:jg1}
\end{eqnarray}
We may consider the set of all feasible $ x $ satisfying $ \dlim_{n \to \infty}\lambda_{n}x_{n+1} = 0 $ in $ ({\rm P}_{1}) $, because of the quadratic minimization and $ \dlim_{n \to \infty}\lambda_{n} = 0. $ This set in turn includes the set of all feasible $ x $ satisfying $ \dlim_{n \to \infty}x_{n} = 0. $ 
Thus letting $n \to \infty $ in (\ref{eqn:jg1}), we have 
\begin{eqnarray}
\dsum_{n=0}^{\infty} \left (x_{n}^{2} + u_{n}^{2}\right) 
\eql c^{2} + c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right] \nonumber \\
& &~ + \dsum_{n=1}^{\infty} \left [x_{n} + \frac{1}{2}(\lambda_{n} - \lambda_{n-1}) \right]^{2} + \dsum_{n=0}^{\infty} \left(u_{n} + \frac{1}{2} \lambda_{n}\right)^{2}.  \label{eqn:gg5}
\end{eqnarray}
%
Similarly, we have for $ (X,\, U) $ 
\begin{eqnarray}
\dsum_{n=0}^{\infty} \left (X_{n}^{2} + U_{n}^{2}\right) 
\eql c^{2} + c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right] \nonumber \\
& &~ + \dsum_{n=1}^{\infty} \left [X_{n} + \frac{1}{2}(\lambda_{n} - \lambda_{n-1}) \right]^{2} + \dsum_{n=0}^{\infty} \left(U_{n} + \frac{1}{2} \lambda_{n}\right)^{2}.  \label{eqn:jh3}
\end{eqnarray}
Since $ (x,\,u) $ satisfies 
 $$ x_{n} + \frac{1}{2}(\lambda_{n} - \lambda_{n-1}) \dql 0,~~~u_{n} + \frac{1}{2} \lambda_{n} \dql 0,~~~\dlim_{n \to \infty}\lambda_{n} \dql 0
 $$
a comparison between (\ref{eqn:gg5}) and (\ref{eqn:jh3}) yields 
\begin{eqnarray*}
\dsum_{n=0}^{\infty} \left (x_{n}^{2} + u_{n}^{2}\right) 
&\le& \dsum_{n=0}^{\infty} \left (X_{n}^{2} + U_{n}^{2}\right). 
\end{eqnarray*}
This completes the proof. \boxend  

From (\ref{eqn:jh3}) we have a basic inequality as follows. 
\begin{lem} It holds that 
\begin{eqnarray}
\dsum_{n=0}^{\infty} \left (x_{n}^{2} + u_{n}^{2}\right) 
\eql c^{2} + c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right] \nonumber \\
& &~ + \dsum_{n=1}^{\infty} \left [x_{n} + \frac{1}{2}(\lambda_{n} - \lambda_{n-1}) \right]^{2} + \dsum_{n=0}^{\infty} \left(u_{n} + \frac{1}{2} \lambda_{n}\right)^{2}  \label{eqn:kg5} \\
%
%
&\ge& c^{2} + c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right]    \nonumber
\end{eqnarray}
for any $ (x,\, u) $ satisfying $ {\rm (i)},\,{\rm (ii)} $ and any $ \lambda $ satisfying $ \dlim_{n \to \infty}\lambda_{n} = 0 $. 
 The equality holds if and only if 
  $$ 2x_{n} = -(\lambda_{n} - \lambda_{n-1})~~ n \ge 1~~~\mbox{and}~~~2u_{n} = -\lambda_{n}~~ n \ge 0. 
 $$
\end{lem}
This lemma states that 
\begin{eqnarray}
&& L(\hat{x}, \hat{u}: \lambda) \le L(\hat{x}, \hat{u}: \lambda^{*}) \le  L(x, u: \lambda^{*}) 
\end{eqnarray}
where 
\begin{eqnarray}
L(x, u: \lambda) 
\eql \dsum_{n=0}^{\infty} \left [x_{n}^{2} + u_{n}^{2} - \lambda_{n}(x_{n+1} - x_{n} - u_{n}) \right]   \nonumber 
\end{eqnarray}
and
\begin{eqnarray}
\hat{x}_{n} \eql c (2 - \phi)^{n}    \nonumber \\[2.56mm]
\hat{u}_{n} \eql -\dfrac{\,c\,}{\,\phi\,} (2 - \phi)^{n}~~~~n \ge 0 \nonumber  \\[2.56mm]
\lambda_{n}^{*} \eql \dfrac{\,2c\,}{\,\phi\,} (2 - \phi)^{n}.   \nonumber 
\end{eqnarray}
In fact, we have the equality between left-hand side and middle side: 
\begin{eqnarray*}
 L(\hat{x}, \hat{u}: \lambda) \eql L(\hat{x}, \hat{u}: \lambda^{*})~~~~~ \forall \lambda ~;~ \dlim_{n \to \infty}\lambda_{n} = 0. 
\end{eqnarray*}

Hence we have a maximization problem for  $ \lambda = (\lambda_{0},\lambda_{1}, \ldots , \lambda_{n}, \ldots) $ as follows:  
%
\begin{eqnarray*}
& &{\rm Maximize} \hspace{3mm} c^{2} + c \lambda_{0} - \frac{1}{4} \dsum_{n=0}^{\infty} \left[ \lambda_{n}^{2} + \left (\lambda_{n} - \lambda_{n+1} \right)^{2}\right]  \\ 
({\rm D}_{1})~~~& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ \lambda \in R^{\infty}  \\[1.05mm]
& &\hspace*{18.5mm}  {\rm (ii)}~~~ \dlim_{n \to \infty}\lambda_{n} = 0.
\end{eqnarray*}
%
Thus we have derived the desired dual problem together with the optimum solution.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Lemma
\begin{lem} 
The point $ \lambda^{*} $ with $ \lambda_{n}^{*} \dql \dfrac{\,2c\,}{\,\phi\,} (2 - \phi)^{n} $ is a maximum point for $({\rm D}_{1})$. It yields the maximum value $ M   = \phi c^{2}  $.  
\end{lem} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Lemma

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Dynamic Programming}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
~

This section discusses a dynamic programming aspect of the two primal problems \cite{bell, bema, iwae, e85}. It is shown that both optimal value function and optimum point function are Golden. The optimal value function is Golden quadratic. The optimum point yields what we call Golden section. 

Let $ f(c) $ be the minimum value of $ ({\rm P}_{1}) $ for $ c \in R^{1}$. Then we have the following results. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{theorem} \label{thm:be1} The minimum value function $ f $ satisfies the Bellman equation
\begin{eqnarray*}
\begin{cases}
f(0) \dql 0  \\[2.14mm]
f(c) \dql \min_{x \in R^{1}} \left[ c^{2} + (c - x)^{2} + f(x) \right].
\end{cases}
\end{eqnarray*}
This has the Golden quadratic minimum value function $ f(c) \dql \phi c^{2} $ for a minimum point function $ \hat{x}(c) \dql \phi^{-2} c. $ 
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem



Let $ g(c) $ be the minimum value of $ ({\rm P}_{2}) $ for $ c \in R^{1}$. Then we have the following results. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Beigin of Theorem
\begin{theorem} \label{thm:be3} The minimum value function $ g $ satisfies the Bellman equation
\begin{eqnarray*}
\begin{cases}
g(0) \dql 0  \\[2.14mm]
g(c) \dql \min_{x \in R^{1}} \left[ (c - x)^{2} + x^{2} + g(x) \right].
\end{cases}
\end{eqnarray*}
This has the inverse-Golden quadratic minimum value function $ g(c) \dql \phi^{-1} c^{2} $ for a minimum point function $ \hat{x}(c) \dql \phi^{-2} c. $ 
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  End of Theorem

We remark that the minimum point function $ \hat{x}(c) \dql \phi^{-2} c $ is called a {\it Golden} decision function. Because it divides an interval $ [0, c] $ into  $ [0, \hat{x}(c)] = [0, (2 - \phi)c] $ and $ [\hat{x}(c), c] = [(2 - \phi)c, c] $, which is the so-called {\it Golden section}. Thus $ \hat{x} $ is Golden (and) optimal. The Golden (Golden optimal) decision function leads to the {\it Golden (Golden optimal) policy} in dynamic programming. Recently the Golden optimal policy for dynamic optimization problems has been discussed in \cite{e78, e80, e88, e87}. It has a strong connection with duality, as we have shown in this paper. 

%\clearpage
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%�i�ȉ��A�Q�l�����𐸑I����I�I�j
%
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%\bibitem{iwj0}��{����: �~���j�A���n�m�V���[�Y �\ �o�ό��ʂƗD���m�� �\, �����\.


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%\bibitem{e78}S.~Iwamoto, Cross dual on the Golden optimum solutions, %�����œK���̌����o��, �����W��u�o�ς̐�����́v�A���吔�����u���^�P�S�S�R�A�Q�O�O�T�N�V���A�����D�Q�V�\�S�R�D

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%�\�P�S�D


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%�u�s�m�����̉��ł̈ӎv����Ɛ������f���v�A���吔�����u���^�P�S�V�V�A�Q�O�O�U�N%%�R���A�����D�P�R�U�\�P�S�O�D


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