248 lines
12 KiB
TeX
248 lines
12 KiB
TeX
\chapter{外文资料原文}
|
|
\label{cha:engorg}
|
|
|
|
\title{The title of the English paper}
|
|
|
|
\textbf{Abstract:} As one of the most widely used techniques in operations
|
|
research, \emph{ mathematical programming} is defined as a means of maximizing a
|
|
quantity known as \emph{bjective function}, subject to a set of constraints
|
|
represented by equations and inequalities. Some known subtopics of mathematical
|
|
programming are linear programming, nonlinear programming, multiobjective
|
|
programming, goal programming, dynamic programming, and multilevel
|
|
programming$^{[1]}$.
|
|
|
|
It is impossible to cover in a single chapter every concept of mathematical
|
|
programming. This chapter introduces only the basic concepts and techniques of
|
|
mathematical programming such that readers gain an understanding of them
|
|
throughout the book$^{[2,3]}$.
|
|
|
|
|
|
\section{Single-Objective Programming}
|
|
The general form of single-objective programming (SOP) is written
|
|
as follows,
|
|
\begin{equation}\tag*{(123)} % 如果附录中的公式不想让它出现在公式索引中，那就请
|
|
% 用 \tag*{xxxx}
|
|
\left\{\begin{array}{l}
|
|
\max \,\,f(x)\\[0.1 cm]
|
|
\mbox{subject to:} \\ [0.1 cm]
|
|
\qquad g_j(x)\le 0,\quad j=1,2,\cdots,p
|
|
\end{array}\right.
|
|
\end{equation}
|
|
which maximizes a real-valued function $f$ of
|
|
$x=(x_1,x_2,\cdots,x_n)$ subject to a set of constraints.
|
|
|
|
\newtheorem{mpdef}{Definition}[chapter]
|
|
\begin{mpdef}
|
|
In SOP, we call $x$ a decision vector, and
|
|
$x_1,x_2,\cdots,x_n$ decision variables. The function
|
|
$f$ is called the objective function. The set
|
|
\begin{equation}\tag*{(456)} % 这里同理，其它不再一一指定。
|
|
S=\left\{x\in\Re^n\bigm|g_j(x)\le 0,\,j=1,2,\cdots,p\right\}
|
|
\end{equation}
|
|
is called the feasible set. An element $x$ in $S$ is called a
|
|
feasible solution.
|
|
\end{mpdef}
|
|
|
|
\newtheorem{mpdefop}[mpdef]{Definition}
|
|
\begin{mpdefop}
|
|
A feasible solution $x^*$ is called the optimal
|
|
solution of SOP if and only if
|
|
\begin{equation}
|
|
f(x^*)\ge f(x)
|
|
\end{equation}
|
|
for any feasible solution $x$.
|
|
\end{mpdefop}
|
|
|
|
One of the outstanding contributions to mathematical programming was known as
|
|
the Kuhn-Tucker conditions\ref{eq:ktc}. In order to introduce them, let us give
|
|
some definitions. An inequality constraint $g_j(x)\le 0$ is said to be active at
|
|
a point $x^*$ if $g_j(x^*)=0$. A point $x^*$ satisfying $g_j(x^*)\le 0$ is said
|
|
to be regular if the gradient vectors $\nabla g_j(x)$ of all active constraints
|
|
are linearly independent.
|
|
|
|
Let $x^*$ be a regular point of the constraints of SOP and assume that all the
|
|
functions $f(x)$ and $g_j(x),j=1,2,\cdots,p$ are differentiable. If $x^*$ is a
|
|
local optimal solution, then there exist Lagrange multipliers
|
|
$\lambda_j,j=1,2,\cdots,p$ such that the following Kuhn-Tucker conditions hold,
|
|
\begin{equation}
|
|
\label{eq:ktc}
|
|
\left\{\begin{array}{l}
|
|
\nabla f(x^*)-\sum\limits_{j=1}^p\lambda_j\nabla g_j(x^*)=0\\[0.3cm]
|
|
\lambda_jg_j(x^*)=0,\quad j=1,2,\cdots,p\\[0.2cm]
|
|
\lambda_j\ge 0,\quad j=1,2,\cdots,p.
|
|
\end{array}\right.
|
|
\end{equation}
|
|
If all the functions $f(x)$ and $g_j(x),j=1,2,\cdots,p$ are convex and
|
|
differentiable, and the point $x^*$ satisfies the Kuhn-Tucker conditions
|
|
(\ref{eq:ktc}), then it has been proved that the point $x^*$ is a global optimal
|
|
solution of SOP.
|
|
|
|
\subsection{Linear Programming}
|
|
\label{sec:lp}
|
|
|
|
If the functions $f(x),g_j(x),j=1,2,\cdots,p$ are all linear, then SOP is called
|
|
a {\em linear programming}.
|
|
|
|
The feasible set of linear is always convex. A point $x$ is called an extreme
|
|
point of convex set $S$ if $x\in S$ and $x$ cannot be expressed as a convex
|
|
combination of two points in $S$. It has been shown that the optimal solution to
|
|
linear programming corresponds to an extreme point of its feasible set provided
|
|
that the feasible set $S$ is bounded. This fact is the basis of the {\em simplex
|
|
algorithm} which was developed by Dantzig as a very efficient method for
|
|
solving linear programming.
|
|
\begin{table}[ht]
|
|
\centering
|
|
\centering
|
|
\caption*{Table~1\hskip1em This is an example for manually numbered table, which
|
|
would not appear in the list of tables}
|
|
\label{tab:badtabular2}
|
|
\begin{tabular}[c]{|m{1.5cm}|c|c|c|c|c|c|}\hline
|
|
\multicolumn{2}{|c|}{Network Topology} & \# of nodes &
|
|
\multicolumn{3}{c|}{\# of clients} & Server \\\hline
|
|
GT-ITM & Waxman Transit-Stub & 600 &
|
|
\multirow{2}{2em}{2\%}&
|
|
\multirow{2}{2em}{10\%}&
|
|
\multirow{2}{2em}{50\%}&
|
|
\multirow{2}{1.2in}{Max. Connectivity}\\\cline{1-3}
|
|
\multicolumn{2}{|c|}{Inet-2.1} & 6000 & & & &\\\hline
|
|
& \multicolumn{2}{c|}{ABCDEF} &\multicolumn{4}{c|}{} \\\hline
|
|
\end{tabular}
|
|
\end{table}
|
|
|
|
Roughly speaking, the simplex algorithm examines only the extreme points of the
|
|
feasible set, rather than all feasible points. At first, the simplex algorithm
|
|
selects an extreme point as the initial point. The successive extreme point is
|
|
selected so as to improve the objective function value. The procedure is
|
|
repeated until no improvement in objective function value can be made. The last
|
|
extreme point is the optimal solution.
|
|
|
|
\subsection{Nonlinear Programming}
|
|
|
|
If at least one of the functions $f(x),g_j(x),j=1,2,\cdots,p$ is nonlinear, then
|
|
SOP is called a {\em nonlinear programming}.
|
|
|
|
A large number of classical optimization methods have been developed to treat
|
|
special-structural nonlinear programming based on the mathematical theory
|
|
concerned with analyzing the structure of problems.
|
|
|
|
Now we consider a nonlinear programming which is confronted solely with
|
|
maximizing a real-valued function with domain $\Re^n$. Whether derivatives are
|
|
available or not, the usual strategy is first to select a point in $\Re^n$ which
|
|
is thought to be the most likely place where the maximum exists. If there is no
|
|
information available on which to base such a selection, a point is chosen at
|
|
random. From this first point an attempt is made to construct a sequence of
|
|
points, each of which yields an improved objective function value over its
|
|
predecessor. The next point to be added to the sequence is chosen by analyzing
|
|
the behavior of the function at the previous points. This construction continues
|
|
until some termination criterion is met. Methods based upon this strategy are
|
|
called {\em ascent methods}, which can be classified as {\em direct methods},
|
|
{\em gradient methods}, and {\em Hessian methods} according to the information
|
|
about the behavior of objective function $f$. Direct methods require only that
|
|
the function can be evaluated at each point. Gradient methods require the
|
|
evaluation of first derivatives of $f$. Hessian methods require the evaluation
|
|
of second derivatives. In fact, there is no superior method for all
|
|
problems. The efficiency of a method is very much dependent upon the objective
|
|
function.
|
|
|
|
\subsection{Integer Programming}
|
|
|
|
{\em Integer programming} is a special mathematical programming in which all of
|
|
the variables are assumed to be only integer values. When there are not only
|
|
integer variables but also conventional continuous variables, we call it {\em
|
|
mixed integer programming}. If all the variables are assumed either 0 or 1,
|
|
then the problem is termed a {\em zero-one programming}. Although integer
|
|
programming can be solved by an {\em exhaustive enumeration} theoretically, it
|
|
is impractical to solve realistically sized integer programming problems. The
|
|
most successful algorithm so far found to solve integer programming is called
|
|
the {\em branch-and-bound enumeration} developed by Balas (1965) and Dakin
|
|
(1965). The other technique to integer programming is the {\em cutting plane
|
|
method} developed by Gomory (1959).
|
|
|
|
\hfill\textit{Uncertain Programming\/}\quad(\textsl{BaoDing Liu, 2006.2})
|
|
|
|
\section*{References}
|
|
\noindent{\itshape NOTE: These references are only for demonstration. They are
|
|
not real citations in the original text.}
|
|
|
|
\begin{translationbib}
|
|
\item Donald E. Knuth. The \TeX book. Addison-Wesley, 1984. ISBN: 0-201-13448-9
|
|
\item Paul W. Abrahams, Karl Berry and Kathryn A. Hargreaves. \TeX\ for the
|
|
Impatient. Addison-Wesley, 1990. ISBN: 0-201-51375-7
|
|
\item David Salomon. The advanced \TeX book. New York : Springer, 1995. ISBN:0-387-94556-3
|
|
\end{translationbib}
|
|
|
|
\chapter{外文资料的调研阅读报告或书面翻译}
|
|
|
|
\title{英文资料的中文标题}
|
|
|
|
{\heiti 摘要：} 本章为外文资料翻译内容。如果有摘要可以直接写上来，这部分好像没有
|
|
明确的规定。
|
|
|
|
\section{单目标规划}
|
|
北冥有鱼，其名为鲲。鲲之大，不知其几千里也。化而为鸟，其名为鹏。鹏之背，不知其几
|
|
千里也。怒而飞，其翼若垂天之云。是鸟也，海运则将徙于南冥。南冥者，天池也。
|
|
\begin{equation}\tag*{(123)}
|
|
p(y|\mathbf{x}) = \frac{p(\mathbf{x},y)}{p(\mathbf{x})}=
|
|
\frac{p(\mathbf{x}|y)p(y)}{p(\mathbf{x})}
|
|
\end{equation}
|
|
|
|
吾生也有涯，而知也无涯。以有涯随无涯，殆已！已而为知者，殆而已矣！为善无近名，为
|
|
恶无近刑，缘督以为经，可以保身，可以全生，可以养亲，可以尽年。
|
|
|
|
\subsection{线性规划}
|
|
庖丁为文惠君解牛，手之所触，肩之所倚，足之所履，膝之所倚，砉然响然，奏刀騞然，莫
|
|
不中音，合于桑林之舞，乃中经首之会。
|
|
\begin{table}[ht]
|
|
\centering
|
|
\centering
|
|
\caption*{表~1\hskip1em 这是手动编号但不出现在索引中的一个表格例子}
|
|
\label{tab:badtabular3}
|
|
\begin{tabular}[c]{|m{1.5cm}|c|c|c|c|c|c|}\hline
|
|
\multicolumn{2}{|c|}{Network Topology} & \# of nodes &
|
|
\multicolumn{3}{c|}{\# of clients} & Server \\\hline
|
|
GT-ITM & Waxman Transit-Stub & 600 &
|
|
\multirow{2}{2em}{2\%}&
|
|
\multirow{2}{2em}{10\%}&
|
|
\multirow{2}{2em}{50\%}&
|
|
\multirow{2}{1.2in}{Max. Connectivity}\\\cline{1-3}
|
|
\multicolumn{2}{|c|}{Inet-2.1} & 6000 & & & &\\\hline
|
|
& \multicolumn{2}{c|}{ABCDEF} &\multicolumn{4}{c|}{} \\\hline
|
|
\end{tabular}
|
|
\end{table}
|
|
|
|
文惠君曰：“嘻，善哉！技盖至此乎？”庖丁释刀对曰：“臣之所好者道也，进乎技矣。始臣之
|
|
解牛之时，所见无非全牛者；三年之后，未尝见全牛也；方今之时，臣以神遇而不以目视，
|
|
官知止而神欲行。依乎天理，批大郤，导大窾，因其固然。技经肯綮之未尝，而况大坬乎！
|
|
良庖岁更刀，割也；族庖月更刀，折也；今臣之刀十九年矣，所解数千牛矣，而刀刃若新发
|
|
于硎。彼节者有间而刀刃者无厚，以无厚入有间，恢恢乎其于游刃必有余地矣。是以十九年
|
|
而刀刃若新发于硎。虽然，每至于族，吾见其难为，怵然为戒，视为止，行为迟，动刀甚微，
|
|
謋然已解，如土委地。提刀而立，为之而四顾，为之踌躇满志，善刀而藏之。”
|
|
|
|
文惠君曰：“善哉！吾闻庖丁之言，得养生焉。”
|
|
|
|
|
|
\subsection{非线性规划}
|
|
孔子与柳下季为友，柳下季之弟名曰盗跖。盗跖从卒九千人，横行天下，侵暴诸侯。穴室枢
|
|
户，驱人牛马，取人妇女。贪得忘亲，不顾父母兄弟，不祭先祖。所过之邑，大国守城，小
|
|
国入保，万民苦之。孔子谓柳下季曰：“夫为人父者，必能诏其子；为人兄者，必能教其弟。
|
|
若父不能诏其子，兄不能教其弟，则无贵父子兄弟之亲矣。今先生，世之才士也，弟为盗
|
|
跖，为天下害，而弗能教也，丘窃为先生羞之。丘请为先生往说之。”
|
|
|
|
柳下季曰：“先生言为人父者必能诏其子，为人兄者必能教其弟，若子不听父之诏，弟不受
|
|
兄之教，虽今先生之辩，将奈之何哉？且跖之为人也，心如涌泉，意如飘风，强足以距敌，
|
|
辩足以饰非。顺其心则喜，逆其心则怒，易辱人以言。先生必无往。”
|
|
|
|
孔子不听，颜回为驭，子贡为右，往见盗跖。
|
|
|
|
\subsection{整数规划}
|
|
盗跖乃方休卒徒大山之阳，脍人肝而餔之。孔子下车而前，见谒者曰：“鲁人孔丘，闻将军
|
|
高义，敬再拜谒者。”谒者入通。盗跖闻之大怒，目如明星，发上指冠，曰：“此夫鲁国之
|
|
巧伪人孔丘非邪？为我告之：尔作言造语，妄称文、武，冠枝木之冠，带死牛之胁，多辞缪
|
|
说，不耕而食，不织而衣，摇唇鼓舌，擅生是非，以迷天下之主，使天下学士不反其本，妄
|
|
作孝弟，而侥幸于封侯富贵者也。子之罪大极重，疾走归！不然，我将以子肝益昼餔之膳。”
|
|
|
|
|
|
\chapter{其它附录}
|
|
前面两个附录主要是给本科生做例子。其它附录的内容可以放到这里，当然如果你愿意，可
|
|
以把这部分也放到独立的文件中，然后将其到主文件中。
|