songsenand
/
cgal
mirror of https://github.com/CGAL/cgal
1
0
Fork 0
Code Issues Packages Projects Releases Wiki Activity

first doc version

This commit is contained in:
Bernd Gärtner 2006-10-17 12:41:03 +00:00
parent 4b42790dbd
commit 968b8cc6f2
2 changed files with 28 additions and 26 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

21
QP_solver/doc_tex/QP_solver_ref/QP_solver.tex
View File

@ -1,9 +1,4 @@
============================================================================= =============================================================================
% The CGAL Reference Manual
% Chapter: Geometric Optimisation
% Section: Smallest Enclosing Sphere
% =============================================================================
\begin{ccRefClass}{QP_solver<Traits>} \begin{ccRefClass}{QP_solver<Traits>}
% ----------------------------------------------------------------------------- % -----------------------------------------------------------------------------
@ -11,19 +6,19 @@
that represents a point $\tilde{x}$ attaining the minimum of a convex that represents a point $\tilde{x}$ attaining the minimum of a convex
quadratic objective function $f(x)=c^{T}x+x^{T}Dx$ on quadratic objective function $f(x)=c^{T}x+x^{T}Dx$ on
a domain $R \subseteq \R^{n}$ given as the intersection $R=\{ x\in\R^n a domain $R \subseteq \R^{n}$ given as the intersection $R=\{ x\in\R^n
\mid \mbox{$Ax\gtreqless b$, $l \le x \le u$}\}$ of halfspaces and \mid \mbox{$Ax\ccTexHtml{\gtreqless}{<=,=,>=} b$, $l \le x \le u$}\}$ of halfspaces and
hyperplanes. In other words, an object of class \ccRefName\ represents hyperplanes. In other words, an object of class \ccRefName\ represents
a solution to the convex mathematical program a solution to the convex mathematical program
\begin{eqnarray*} \begin{eqnarray*}
\mbox{(QP)}&\mbox{minimize} & c^{T}x+x^{T}Dx \\ \mbox{(QP)}&\mbox{minimize} & c^{T}x+x^{T}Dx \\
&\mbox{subject to} & Ax \gtreqless b, \\ &\mbox{subject to} & Ax\ccTexHtml{\gtreqless}{<=,=,>=} b, \\
&& l \leq x \leq u, && l \leq x \leq u,
\end{eqnarray*} \end{eqnarray*}
over the $n$ variables $x=(x_0,\ldots,x_{n-1})$. over the $n$ variables $x=(x_0,\ldots,x_{n-1})$.
Here, $A \in \R^{m \times n}$, $b \in \R^{m}$, $c \in \R^{n}$, and in Here, $A \in \R^{m \times n}$, $b \in \R^{m}$, $c \in \R^{n}$, and in
addition, the matrix $D \in \R^{n \times n}$ is positive-semidefinite. addition, the matrix $D \in \R^{n \times n}$ is positive-semidefinite.
The symbol ``$\gtreqless$'' in the program's constraint $Ax \gtreqless The symbol ``$\ccTexHtml{\gtreqless}{<=,=,>=}$'' in the program's constraint $Ax \ccTexHtml{\gtreqless}{<=,=,>=}
b$ means that any row of this may either be an inequality $(Ax)_i \leq b_i$ or b$ means that any row of this may either be an inequality $(Ax)_i \leq b_i$ or
$(Ax)_i \geq b_i$, or an equality $(Ax)_i = b_i$. Furthermore, $l\in $(Ax)_i \geq b_i$, or an equality $(Ax)_i = b_i$. Furthermore, $l\in
(\R\cup\{-\infty\})^n$ and $r\in (\R\cup\{\infty\})^n$, so that each (\R\cup\{-\infty\})^n$ and $r\in (\R\cup\{\infty\})^n$, so that each
@ -66,7 +61,7 @@ variables are zero (and therefore need not be queried).
\paragraph{Usage.} \paragraph{Usage.}
An object of class \ccRefName\ is constructed by specifing $A, b, c, An object of class \ccRefName\ is constructed by specifing $A, b, c,
D, l, u$, and for each constraint from $Ax \gtreqless b$ whether it D, l, u$, and for each constraint from $Ax \ccTexHtml{\gtreqless}{<=,=,>=} b$ whether it
needs to hold with inequality ($\leq$ or $\geq$) or with equality. needs to hold with inequality ($\leq$ or $\geq$) or with equality.
The algorithm then determines the object's \emph{status}, which The algorithm then determines the object's \emph{status}, which
reflects whether the quadratic program (QP) is infeasible, unbounded, reflects whether the quadratic program (QP) is infeasible, unbounded,
@ -119,7 +114,7 @@ In order to improve performance, the algorithm can be tailored to
specific variants of the quadratic program. In particular, if the specific variants of the quadratic program. In particular, if the
program is (at compile time) known to be linear (i.e., $D=0$), or in program is (at compile time) known to be linear (i.e., $D=0$), or in
standard form, or symmetric (meaning $D^{T}=D$), or if the constraint standard form, or symmetric (meaning $D^{T}=D$), or if the constraint
matrix $A$ is known to have full row rank and $Ax \gtreqless b$ matrix $A$ is known to have full row rank and $Ax \ccTexHtml{\gtreqless}{<=,=,>=} b$
consists of equalities only then compile-time flags in the traits consists of equalities only then compile-time flags in the traits
class \ccc{Traits} can be used to specialize the algorithm. The class \ccc{Traits} can be used to specialize the algorithm. The
resulting code will take advantage of the given additional knowledge resulting code will take advantage of the given additional knowledge
@ -292,7 +287,7 @@ all entries of $u$ are $\infty$ (i.e., (QP) is assumed to be in
standard form). The matrix $A$ of (QP) is given by \ccc{a_it}, which standard form). The matrix $A$ of (QP) is given by \ccc{a_it}, which
is an iterator over the columns of $A$. Similarly, \ccc{d_it} is an iterator over the columns of $A$. Similarly, \ccc{d_it}
iterates over the rows of $D$. The iterator \ccc{r_it} specifies for iterates over the rows of $D$. The iterator \ccc{r_it} specifies for
each row of $Ax \gtreqless b$ whether it is an inequality ($\leq$ or each row of $Ax \ccTexHtml{\gtreqless}{<=,=,>=} b$ whether it is an inequality ($\leq$ or
$\geq$) or an equality constraint. \ccRequire $\geq$) or an equality constraint. \ccRequire
\ccc{a_it} iterates over $n$ columns each of which has $m$ elements, \ccc{a_it} iterates over $n$ columns each of which has $m$ elements,
\ccc{d_it} iterates over $n$ rows each of which having $n$ elements, \ccc{d_it} iterates over $n$ rows each of which having $n$ elements,
@ -309,7 +304,7 @@ strategy = FULL_EXACT_PRICING)}
program (QP). The matrix $A$ of (QP) is given by \ccc{a_it}, which is program (QP). The matrix $A$ of (QP) is given by \ccc{a_it}, which is
an iterator over the columns of $A$. Similarly, \ccc{d_it} iterates an iterator over the columns of $A$. Similarly, \ccc{d_it} iterates
over the rows of $D$. The iterator \ccc{r_it} specifies for each row over the rows of $D$. The iterator \ccc{r_it} specifies for each row
of $Ax \gtreqless b$ whether it is an inequality ($\leq$ or $\geq$) or of $Ax \ccTexHtml{\gtreqless}{<=,=,>=} b$ whether it is an inequality ($\leq$ or $\geq$) or
an equality constraint. \ccc{fl_it} iterates over $n$ Boolean values, an equality constraint. \ccc{fl_it} iterates over $n$ Boolean values,
one for each variable $x_i$, $i\in\{0,\ldots,n-1\}$, and specifies one for each variable $x_i$, $i\in\{0,\ldots,n-1\}$, and specifies
whether $l_i>-\infty$. If so, the corresponding entry from the whether $l_i>-\infty$. If so, the corresponding entry from the
@ -541,7 +536,7 @@ The following example solves the quadratic programming problem
&& 0 \le x_1 \le 1. && 0 \le x_1 \le 1.
\end{eqnarray*} \end{eqnarray*}
\ccIncludeExampleCode{QP_solver/small_example.cpp} \ccIncludeExampleCode{QP_solver/qp_solver1.cpp}
Please refer also to the documentation of class Please refer also to the documentation of class
\ccc{CGAL::QP_MPS_instance<IT,ET>}; using it, you can read programs of \ccc{CGAL::QP_MPS_instance<IT,ET>}; using it, you can read programs of

33
QP_solver/doc_tex/QP_solver_ref/main.tex
View File

@ -3,18 +3,25 @@
% Chapter: Geometric Optimisation % Chapter: Geometric Optimisation
% Section: QP solver % Section: QP solver
% ============================================================================= % =============================================================================
\newcommand{\qprel}{\ccTexHtml{\gtreqless}{&nbsp;~&nbsp;}}
\input{Optimisation_ref/QP_solver} \input{QP_solver_ref/QuadraticProgramInterface}
\input{Optimisation_ref/Traits} \input{QP_solver_ref/NonnegativeQuadraticProgramInterface}
\input{Optimisation_ref/MPS} \input{QP_solver_ref/LinearProgramInterface}
\input{Optimisation_ref/QP_solver_MPS_traits_d} \input{QP_solver_ref/NonnegativeLinearProgramInterface}
%\input{Optimisation_ref/QP_pricing_strategy}
%\input{Optimisation_ref/QP__filtered_base} \input{QP_solver_ref/Quadratic_program_from_iterators}
%\input{Optimisation_ref/QP_full_exact_pricing} %\input{QP_solver_ref/QP_solver}
%\input{Optimisation_ref/QP_full_filtered_pricing} %\input{QP_solver_ref/Traits}
%\input{Optimisation_ref/QP_partial_exact_pricing} %\input{QP_solver_ref/MPS}
%\input{Optimisation_ref/QP_partial_filtered_pricing} %\input{QP_solver_ref/QP_solver_MPS_traits_d}
%\input{QP_solver_ref/QP_pricing_strategy}
%\input{QP_solver_ref/QP__filtered_base}
%\input{QP_solver_ref/QP_full_exact_pricing}
%\input{QP_solver_ref/QP_full_filtered_pricing}
%\input{QP_solver_ref/QP_partial_exact_pricing}
%\input{QP_solver_ref/QP_partial_filtered_pricing}
% If you have models for your concepts, include them here: % If you have models for your concepts, include them here:
%\input{Optimisation_ref/Traits_model_2} %\input{QP_solver_ref/Traits_model_2}
%\input{Optimisation_ref/Traits_model_3} %\input{QP_solver_ref/Traits_model_3}
%\input{Optimisation_ref/Traits_model_d} %\input{QP_solver_ref/Traits_model_d}