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\title{QP\_solver}
\author{}
%
\begin{document}
\maketitle
\begin{slide}
\emph{QP:}
\begin{eqnarray*}
minimize & c^{T}x+x^{T}Dx & \\
s.t.     & Ax \leq b & \\
         & x \geq 0 ,&
\end{eqnarray*}
where
$A \in \R^{m \times n}$, $b \in \R^{m}$, $c \in \R^{n}$ and $D \in \R^{n
\times n}$ positive-semidefinite.


\end{slide}

\begin{note}
\begin{itemize}
\item linear part of objective function: $c$
\item quadratic part of objective function: $D$
\item constraint matrix: $A$
\begin{itemize}
\item $n$ variables
\item $m$ constraints
\end{itemize}
\item all of the variables are required to be greater equal zero.
\item The solver is tailored to quadratic programs where
$\min\{n,m\}$ is small.
\item an example where this is the case is POLYTOPE DISTANCE
\end{itemize}
\end{note}

\begin{slide}
\emph{Polytope distance}

Given two point sets in $d$-dimensional space,
\[
P=\{p_{1}, \ldots, p_{r} \},
\quad
Q=\{q_{1}, \ldots, q_{s} \}
\]
with $r+s=n$. Let
\[
C=(p_{1}, \ldots , p_{r}, -q_{1}, \ldots, -q_{s}).
\]
Then 
\begin{eqnarray*}
(PD) & minimize & x^{T}C^{T}Cx  \\
     & s.t.     & \sum_{i=1}^{r}x_{i} = 1 \\
     &          & \sum_{i=r+1}^{n}x_{i} = 1 \\
     &          & x \geq 0,
\end{eqnarray*}
minimizes the square of the euclidean distance between $conv(P)$ and $conv(Q)$.
\end{slide}

\begin{note}
\begin{itemize}
\item $Cx$ is the difference vector of some convex combinations over the point
sets $P$, $Q$.
\item Note, here $\min\{n,m\}=2$
\end{itemize}
\end{note}

\begin{slide}
\emph{Features:}
\begin{itemize}
\item hybrid arithmetic: ($ET$, $NT$)
\item configurability for specific situations:
  \begin{enumerate}
  \item compile time tags: 
%  \begin{itemize}
%   \item Is\_linear
%   \item Is\_symmetric
%   \item Has\_full\_row\_rank
%   \item Use\_perturbation
%  \end{itemize}
  \item Pricing strategies
  \end{enumerate}
\item symbolic perturbation
\item handling of implicitly given matrices
\end{itemize}
\end{slide}

\begin{note}
\begin{itemize}
\item \emph{hybrid arithmetic}: exact number type and fast, usually inexact
number type $NT$ are used by the solver. The fast number type is used in
connection with some pricing strategies.
\item \emph{configurability}: different compile time tags, most of them
improving performance for specific optimization problems.$\rightarrow$ example
\item \emph{pricing strategies}: $4$ ready-to-use pricing strategies are
provided, which can be specified in the solvers constructor.
\item \emph{symbolic perturbation}: guarantees termination.  
There are indications that cycling occurs more often with exact arithmetic as
one would expect, for most of the times it goes unnoticed with inexact
arithmetic.
\item \emph{implicitly given matrices}: $\rightarrow$ example. 
\end{itemize}
\end{note}



\begin{slide}
Example:

\begin{eqnarray*}
minimize & c^{T}x+x^{T}Dx & \\
s.t.     & Ax \geq b & \\
         & x \geq 0  ,&
\end{eqnarray*}
where
\[
A:=
\left(
\begin{array}{rrr}
-4 & 2 & 0 \\
1  & 1 & 1 \\
\end{array}
\right),
\quad
b:=
\left(
\begin{array}{r}
-8 \\
2
\end{array}
\right),
\]
\[
c^{T}:=
\left(
\begin{array}{rrr}
0 & 5 & 0
\end{array}
\right),
\quad
D:=
\left(
\begin{array}{rrr}
64  & -16 & 0 \\
-16 &   4 & 2 \\
0   &  -2 & 0 
\end{array}
\right).
\]
\end{slide}


\begin{note}
\begin{itemize}
\item  matrix $A$ has full row rank
\item  matrix $D$ is not symmetric
\item all constraints are greater equal constraints 
\end{itemize}
\end{note}

\begin{slide}
QP\_solver\_talk\_example.cpp first part
\end{slide}

\begin{note}
\begin{itemize}
\item exact number type $ET$.
\item \emph{A\_iterator, D\_iterator}: model of the STL-concept
RandomAccessIterator referring to an iterator, which in turn refers to the
elements of a column/row of the matrix $A/D$.
\item \emph{B\_iterator}, \emph{C\_iterator}: model of the STL-concept
RandomAccessIterator referring to the entries of $b$ and $c$.
\item \emph{Row\_type\_iterator}: model of the STL-concept
RandomAccessIterator referring to the constraint types of the rows of constr.
matrix $A$.  
\item element types of $A$, $b$, $c$ and $D$ must provide implicit conversions
to $ET$.
\item \emph{QP\_const\_value\_iterator} convenience class template
\end{itemize}
\end{note}

\begin{slide}
QP\_solver\_talk\_example.cpp second part
\end{slide}

\begin{note}
\begin{itemize}
\item show how columns of $A$ and rows of $D$ are inserted.
\item \emph{QP\_solver$<$QPSolverTraits$>$} constructor defaulting to
\emph{full exact pricing} strategy
\item QP\_solver solution query: qp.status()
\item Variable\_value\_iterator: qp.variables\_value\_begin(),
qp.variables\_value\_end()
\end{itemize}
\end{note}

%
%\begin{slide}
%\begin{itemize}
%\item The solver is tailored to quadratic programs where
%$\min\{n,m\}$ is small.
%\end{itemize}
%\end{slide}

%\begin{note}
%\begin{itemize}
%\item $QP$-bases are generalizations of $LP$-bases, coincide if $D=0$
%\item computations with basis inverses exact $\rightarrow$ slow
%\item ok, as long as $\min\{m,n\}$ is small
%\item and pricing is fast
%\end{itemize}
%\end{note}


\begin{slide}
\emph{compile time tags:}
\begin{itemize}
\item \emph{Is\_linear}: specifies whether the problem to be solved is a linear
program or a quadratic program.
\item \emph{Is\_symmetric}: specifies whether the matrix $D$ is symmetric.  
\item \emph{Has\_full\_row\_rank}: specifies whether the constraint matrix $A$
has full row rank.
\item \emph{Use\_perturbation}: specifies whether symbolic perturbation is 
used to guarantee termination.
\end{itemize}
\end{slide}

\begin{note}
\begin{itemize}
\item almost no overhead incurred for the linear case
\item $D$ symmetric halves accesses to the matrix $D$
\item Enhances performance, if constraint matrix $A$ has full row rank
\item Avoids cycling at the cost of a small performance penalty 
\end{itemize}
\end{note}

\begin{slide}
\begin{itemize}
\item The solver uses an \emph{iterative algorithm}, each iteration yields a
\emph{feasible solution} $x$, which is characterized by a subset
$B \subseteq \left[n\right]$, such that $x_{B} \geq 0$, variables not in $B$
are \emph{nonbasic}. Nonbasic variables have value zero.   

\item The \emph{pricing step} of the solvers algorithm
determines whether the current solution is optimal;
if not, an \emph{entering variable} among the nonbasic variables is determined.

\item A nonbasic variable $x_{j}$ \emph{qualifies} as entering variable,
if some associated value $\mu_{j}$ is strictly negative. If none of the
nonbasic variables qualifies as entering variable the current solution is
\emph{optimal}.

\item The \emph{pricing strategy} is the part of the algorithms
pricing step that selects an entering variable among the
nonbasic variables that qualify as entering variable.
\end{itemize}
\end{slide}

\begin{note}
\begin{itemize}
\item mention algorithm used is actually a generalization of LP-simplex
\item for the linear case the notions of $\mu_{j}$ and reduced cost coincide
\end{itemize}
\end{note}

\begin{slide}
Four ready-to-use pricing strategies are provided:
\begin{itemize}
\item \emph{full exact pricing} 
\item \emph{full filtered pricing}
\item \emph{partial exact pricing}
\item \emph{partial filtered pricing}
\end{itemize}
User provided pricing strategies may be specified by means of inheritance.
\end{slide}

\begin{note}
\begin{itemize}
\item User provided pricing strategies inherit from the base class
\begin{verbatim}
QP_pricing_strategy<Traits>
\end{verbatim}

\item issue: should this base class be made public?
\item currently used as developers feature
\item interface between solver and the pricing strategy becomes fat
\end{itemize}
\end{note}


\begin{slide}
\emph{Full exact pricing}

The $\mu_{j}$ associated with the nonbasic $x_{j}$, $j \in N$, are
evaluated using the exact number type $ET$.

\emph{Creation:}
\begin{verbatim}
QP_full_exact_pricing<Traits> feps();
\end{verbatim}
\end{slide}

\begin{note}
\begin{itemize}
\item is the default pricing strategy
\end{itemize}
\end{note}

\begin{slide}
\emph{Full filtered pricing}

\begin{itemize}
\item Approximations $\tilde{\mu}_{j}$, $j \in N$, are computed using the fast
(compared to $ET$) number type $NT$ at beginning of the pricing step.

\item If $\mu_{j_{N}} < 0$, $j_{N}=\arg\min_{j \in N}\tilde{\mu}_{j}$, then 
$\mu_{j_{N}}$ is the entering variable.

\item Otherwise, \emph{nonnegativity} for part of the nonbasic variables is
\emph{certified} using an error bound $b$ on
$\left|\tilde{\mu}_{j}-\mu_{j}\right|$: if $\tilde{\mu}_{j} \geq b$ then
$\mu_{j}$ is known to be nonnegative.

\item The $\tilde{\mu}_{j}$, $j \in N$ whose exact $\mu_{j}$ values cannot be
certified to be nonnegative by this \emph{floating point filter} are then
evaluated using exact arithmetic. 
\end{itemize}
\end{slide}

\begin{note}
\begin{itemize}
\item One of the two pricing strategies that uses hybrid arithmetic,
basic idea: use (fast) approx for $\mu_{j}$ and an error bound
\item Since $\mu_{j}$ is basically a scalar product, one can estimate the 
absolute value of the diff between $\mu_{j}$ and its approximation by a bound
$b$. If the approximated value is larger than bound $b$, $\mu_{j}$ is
\emph{certified} to be nonnegative
\item (slide): the minimal value of the approximated $\mu_{j}$ is chosen because
it has highest prob. to be strictly negative.
\item Note, the error bound consists actually of two bounds, one
independent of $j$ and one depending on $j$, bound $b$ can be evaluated
using the fast (inexact) number type $NT$.
\item improves performance if number type $NT$ is fast compared to $ET$
\end{itemize}
\end{note}


\begin{slide}
\emph{Full filtered pricing}

\emph{Creation:}
\begin{verbatim}
QP_full_filtered_pricing<Traits, NT, ET2NT>
  ffps( ET2NT et2nt = ET2NT());
\end{verbatim}
\end{slide}

\begin{note}
\begin{itemize}
\item The input types of $A$, $b$, $c$, as well as $D$ are required to provide
implicit lossless conversion to $NT$.
\item Currently $NT$ is supported only for $double$.
\end{itemize}
\end{note}

\begin{slide}
\emph{Partial exact pricing}

\begin{itemize}
\item A set $S$ of \emph{active variables}, $S \subseteq N$, is kept
and an entering variable is chosen among the variables in $S$ that
qualify as entering variable.

\item If none of the variables in $S$ qualifies as entering variable
the set $S$ is augmented by
$V:=\{k \in N \setminus S \left|\right. \mu_{k} < 0\}$.

\item Initially, $\left|S\right|=\min\{m\sqrt{n/2}, n\}$.
\end{itemize}


\emph{Creation:}
\begin{verbatim}
QP_partial_exact_pricing<Traits> peps(
  bool randomize = false, 
  Random& random = default_random);
\end{verbatim}
\end{slide}

\begin{note}
\begin{itemize}
\item intuition of \emph{partial pricing}: to maintain a set of active
variables, which is initially relatively small and to find the entering
variable in the set of active variables.
\item first try the set of \emph{active variables}, if this fails try the
complement $S$ with respect to the nonbasic variables.
\item (slide) $S:=S \cup V$ for next iteration. 
\item constructor and choices for the initial set, either as the first
$\min\{m\sqrt{n/2},n\}$ nonbasic variables determined by the initial feasible
solution or chosen randomly out of the nonbasic variables determined by the
initial feasible solution. 
\item \emph{partial exact pricing} improves performance for optimization
problems with high \emph{variables-to-constraint} ratios.
\end{itemize}
\end{note}


\begin{slide}
\emph{Partial filtered pricing}

\begin{itemize}
\item If $\mu_{j_{S}} < 0$, $j_{S}:=\arg\min_{j \in S}\tilde{\mu}_{j}$, then
$x_{j_{S}}$ is the entering variable.

\item Otherwise, $V:=\{k \in N \setminus S \left|\right. \tilde{\mu}_{j} <0\}$
is determined. If $V \neq \emptyset$ and $\mu_{j_{V}} <0$,
$j_{V}:=\arg\min_{j \in V}\tilde{\mu}_{j}$, then $x_{j_{V}}$ is the entering
variable.

\item If $V=\emptyset$, that is
$\tilde{\mu}_{j} \geq 0$ for $k \in N \setminus S$, or $\mu_{j_{V}} \geq 0$
the nonnegativity for
$\mu_{j}$, $j \in N$ is
either certified using the error bound, $\tilde{\mu}_{j} \geq b$,
or the exact value $\mu_{j}$ is evaluated.
\end{itemize}

\emph{Creation:}
\begin{verbatim}
QP_partial_filtered_pricing<Traits, NT, ET2NT>
  pfps( ET2NT et2nt = ET2NT());
\end{verbatim}
\end{slide}

\begin{note}
\begin{itemize}
\item \emph{partial filtered pricing} is the combination of partial pricing
and filtered pricing
\item As with partial exact pricing we have a set of \emph{active variables} and
as with full filtered pricing we use approx. and try to certify noneg. using
the floating point filter $\rightarrow$ slide
\item Again as with full filtered pricing the input types of $A$, $b$, $c$,
as well as $D$ are required to provide implicit lossless conversion to $NT$.
Currently $NT$ is supported only for $double$. 
\item improves performance of QP with high \emph{variables-to-constraints}
ratio and number types $NT$ that are fast compared to the number type $ET$.
\end{itemize}
\end{note}


\begin{slide}
\emph{Polytope distance}

Given two point sets in $d$-dimensional space,
\[
P=\{p_{1}, \ldots, p_{r} \},
\quad
Q=\{q_{1}, \ldots, q_{s} \}
\]
with $r+s=n$. Let
\[
C=(p_{1}, \ldots , p_{r}, -q_{1}, \ldots, -q_{s}).
\]
Then 
\begin{eqnarray*}
(PD) & minimize & x^{T}C^{T}Cx  \\
     & s.t.     & \sum_{i=1}^{r}x_{i} = 1 \\
     &          & \sum_{i=r+1}^{n}x_{i} = 1 \\
     &          & x \geq 0,
\end{eqnarray*}
minimizes the square of the euclidean distance between $conv(P)$ and $conv(Q)$.
Here, $D=C^{T}C$ is an
$n \times n$-matrix, but its rank is only $d$.
\end{slide}

\begin{note}
\begin{itemize}
\item $c$ is $0$-vector
\item $b$ is const vector
\item row\_types are all equality constraints
\item $D$ is $n \times n$ matrix, given in terms of the $d \times n$
matrix $C$.  
\end{itemize}
\end{note}

\begin{slide}
example PD\_D\_iterator
\end{slide}

\begin{note}
\begin{itemize}
\item start with constructor
\begin{enumerate}
\item point\_it: iterator referring to $C$ matrix
\item i current row index
\end{enumerate}
\item dereference operator, returns a PD\_D\_row\_iterator
\end{itemize}
\end{note}

\begin{slide}
example PD\_D\_row\_iterator
\end{slide}

\begin{note}
\begin{itemize}
\item constructor: point\_it referring to point matrix $C$
\item dereference operator returns scalar product of columns $i$ and $j$ of the
$C$ matrix.
\end{itemize}
\end{note}

\begin{slide}
\begin{theorem}(G\"{a}rtner)
Every QP-basis $B$ satisfies
\[
\left|B\right| \leq m + Rank(D).
\]
\end{theorem}
\end{slide}

\begin{note}
\begin{itemize}
\item in general our $B$ and QP-basis $B$ are different, but in this particular
instance they coincide
\item thus here $\left|B\right| \leq 2 + d$.
\end{itemize}
\end{note}


%\begin{slide}
%Applications:
%\begin{itemize}
%\item solver is suitable for problems with $\min\{n,m\}$ small, in contrast to
%the \emph{operations research} setting, where $m$ and $n$ are large and
%$A$ and $D$ \emph{sparse} 
%\item implicitly given matrices
%\item Polytope Distance (primal version) example
%\end{itemize}
%\end{slide}
\end{document}
