mirror of https://github.com/CGAL/cgal
136 lines
4.9 KiB
TeX
136 lines
4.9 KiB
TeX
\begin{ccRefConcept}{QuadraticProgramInterface}
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\ccDefinition
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A model of \ccRefName\ describes a convex quadratic program of the form
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%%
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\begin{eqnarray*}
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\mbox{(QP)}& \mbox{minimize} & x^{T}Dx+c^{T}x+c_0 \\
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&\mbox{subject to} & Ax\qprel b, \\
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& & l \leq x \leq u
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\end{eqnarray*}
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%%
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in $n$ real variables $x=(x_0,\ldots,x_{n-1})$.
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Here,
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\begin{itemize}
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\item $A$ is an $m\times n$ matrix (the constraint matrix),
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\item $b$ is an $m$-dimensional vector (the right-hand side),
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\item $\qprel$ is an $m$-dimensional vector of relations
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from $\{\leq, =, \geq\}$,
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\item $l$ is an $n$-dimensional vector of lower
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bounds for $x$,
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\item $u$ is an $n$-dimensional vector of upper bounds for
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$x$,
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\item $D$ is a symmetric positive-semidefinite $n\times n$ matrix (the
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quadratic objective function),
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\item $c$ is an $n$-dimensional vector (the linear objective
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function), and
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\item $c_0$ is a constant.
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\end{itemize}
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\ccRefines
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\ccc{NonnegativeQuadraticProgramInterface}\\
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\ccc{LinearProgramInterface}\\
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\ccc{NonnegativeLinearProgramInterface}
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\ccHasModels
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\ccc{CGAL::Quadratic_program_from_iterators<A_it, B_it, R_it, FL_it, L_it, FU_it, U_it, D_it, C_it>}\\
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\ccc{CGAL::Quadratic_program_from_pointers<NT>}\\
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\ccc{CGAL::Quadratic_program_from_mps<NT>}
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\ccTypes
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\ccNestedType{A_iterator}{A random access iterator type for
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the \emph{columns} of the constraint matrix $A$.}
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\ccNestedType{B_iterator}{A random access iterator type for
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the entries of the right-hand side $b$.}
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\ccNestedType{R_iterator}{A random access iterator type for the
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relations $\qprel$. The value type of \ccc{R_iterator} is
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\ccc{CGAL::Comparison_result}.}
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\ccNestedType{FL_iterator}{A random access iterator type for the
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existence (or finiteness) of the lower bounds $l_j, j=0,\ldots,n-1$.
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The value type of \ccc{FL_iterator} is \ccc{bool}.}
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\ccNestedType{L_iterator}{A random acess iterator type for the entries
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of the lower bound vector $l$.}
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\ccNestedType{UL_iterator}{A random access iterator type for the
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existence (or finiteness) of the upper bounds $u_j, j=0,\ldots,n-1$.
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The value type of \ccc{UL_iterator} is \ccc{bool}.}
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\ccNestedType{U_iterator}{A random acess iterator type for the entries
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of the upper bound vector $u$.}
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\ccNestedType{D_iterator}{A random access iterator type for the rows
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of the quadratic objective function matrix $D$.}
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\ccNestedType{C_iterator}{A random access iterator type for the
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entries of the linear objective function vector $c$.}
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\ccOperations
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\ccCreationVariable{qp}
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\ccMethod{int n() const;}{returns the number $n$ of variables (number
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of columns of $A$) in \ccVar.}
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\ccMethod{int m() const;}{returns the number $m$ of constraints
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(number of rows of $A$) in \ccVar.}
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\ccMethod{const A_iterator& a() const;}{returns an iterator for the columns
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of $A$. For $j=0,\ldots,n-1$, $\ccVar.\ccc{a()}[j]$ is a random access
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iterator for column $j$. This means that $\ccVar.\ccc{a()}[j][i]$ is the
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entry of $A$ in row $i$ and column $j$ (row and column indices start
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from $0$).}
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\ccMethod{const B_iterator& b() const;}{returns an iterator for the entries
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of $b$.}
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\ccMethod{const R_iterator& r() const;}{returns an iterator for the entries
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of $\qprel$. The value \ccc{CGAL::SMALLER} stands
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for $\leq$, \ccc{CGAL::EQUAL} stands for $=$, and \ccc{CGAL::LARGER}
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stands for $\geq$.}
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\ccMethod{const FL_iterator& fl() const;}{returns an iterator for the
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existence of the lower bounds $l_j, j=0,\ldots,n-1$. If
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$\ccVar.\ccc{fl()}[j]=true$, the variable $x_j$ has a lower
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bound, otherwise it has no lower bound.}
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\ccMethod{const L_iterator& l() const;}{returns an iterator for the
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entries of $l$. If $\ccVar.\ccc{fl()}[j]=\ccc{false}$, the value
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$\ccVar.\ccc{l()}[j]$ is not accessed.}
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\ccMethod{const FU_iterator& fu() const;}{returns an iterator for the
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existence of the upper bounds $u_j, j=0,\ldots,n-1$. If
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$\ccVar.\ccc{fu()}[j]=true$, the variable $x_j$ has an upper
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bound, otherwise it has no upper bound.}
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\ccMethod{const U_iterator& u() const;}{returns an iterator for the
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entries of $u$. If $\ccVar.\ccc{fu()}[j]=\ccc{false}$, the value
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$\ccVar.\ccc{u()}[j]$ is not accessed.}
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\ccMethod{const D_iterator& d() const;}{returns an iterator for the rows of
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$D$. For $i=0,\ldots,n-1$, $\ccVar.\ccc{d()}[i]$ is a random access
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iterator for the entries in row $i$ \emph{below or on the diagonal}.
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This means that $\ccVar.\ccc{d()}[i][j]$ is valid only for $j\leq i$
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and returns the entry of $D$ in row $i$ and column $j$ (row and
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column indices start from $0$).}
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\ccMethod{const C_iterator& c() const;}{returns an iterator for the entries
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of $c$.}
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\ccMethod{const C_iterator::value_type& c0() const;}{returns the constant term
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$c_0$ of the objective function.}
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\ccRequirements
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The value types of all iterator types (nested iterator types,
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respectively, for \ccc{A_iterator} and \ccc{D_iterator}) must be
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convertible to some common Euclidian ring number type \ccc{ET}.
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\ccSeeAlso
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\end{ccRefConcept}
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