# SCM Repository

 [directlabels] / tex / useR-2011 / quadprog.Rnw

Mon Aug 8 14:35:53 2011 UTC (7 years, 5 months ago) by tdhock
File size: 2453 byte(s)
changes from karteek
\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{amssymb,amsmath}
\usepackage{tikz}
\usepackage[nogin]{Sweave}
\newcommand{\RR}{\mathbb R}
\pagestyle{empty}
\begin{document}
Let $t_1\leq \dots\leq t_k$ be the target locations for each of the
$k$ direct labels, and let $h_1, ..., h_k$ be the heights of the
corresponding labels.

<<fig=TRUE,tikz=TRUE,echo=FALSE,width=6,height=3,results=hide>>=
par(mar=c(0,0,0,3))
y <- c(1,2,5,6)
x <- rep(0,length(y))
h <- c(1,0.5,1,0.8)
plot(c(0,0.5),c(0,7),type="n")
rect(0,y-h/2,0.5,y+h/2,col="grey90")
i <- c(1,2,"k-1","k")
axis(4,y,sprintf("$t_{%s}$",i),las=2)
axis(4,3.5,"$\\vdots$",las=2,tick=FALSE)
arrows(0.2,y-h/2,0.2,y+h/2,0.1,code=3)
text(0.25,y,sprintf("$h_{%s}$",i),adj=c(0,0.5))
text(0.25,3.5,"$\\vdots$",adj=c(0,0.5))
@

The optimal direct labels do not overlap, and are as close as
possible to the target locations:
\begin{aligned} &\min_{b\in\RR^k} && \sum_{i=1}^k (b_i-t_i)^2 =||b-t||^2 \\ &\text{subject to} && b_{i+1} \geq b_i + h_{i+1}/2+h_i/2, \ \forall\ i=1,...,k-1 \end{aligned}
This is a quadratic program (QP) that we can solve using
\texttt{quadprog::solve.QP()} and we can use the optimal $b$ for the
direct label positions. To use the solver, we must write the QP in
standard form:
\label{eq:standard} \begin{aligned} &\min_{b\in\RR^k} && \frac 1 2 b^\prime b - t^\prime b\\ &\text{subject to} && A^\prime b = \left[ \begin{array}{cccccc} -1 & 1 & 0 \\ 0 & -1 & 1\\ &&\ddots&\ddots\\ &&&-1&1&0\\ &&&0&-1&1 \end{array} \right] \left[ \begin{array}{c} b_1\\ \vdots\\ b_k \end{array} \right] \geq \underbrace{ \left[ \begin{array}{c} (h_1 + h_2)/2\\ \vdots\\ (h_{k-1}+h_k)/2 \end{array} \right] }_{\texttt{(h[-k]+h[-1])/2}} \end{aligned}
where $A$ is the $k\times k-1$ constraint coefficient matrix:
$$\label{eq:A} A= \left[ \begin{array}{ccccc} -1 & 0 \\ 1 & -1\\ 0 & 1 & \ddots\\ &&\ddots&-1&0\\ &&&1&-1\\ &&&0&1 \end{array} \right] = \underbrace{ \left[ \begin{array}{ccc} 0 & \cdots & 0\\ 1 & & 0\\ & \ddots\\ 0 & & 1 \end{array} \right] }_{\texttt{rbind(0,I_{k-1})}} - \underbrace{ \left[ \begin{array}{ccc} 1 & & 0\\ & \ddots\\ 0 & & 1\\ 0 & \cdots & 0 \end{array} \right] }_{\texttt{rbind(I_{k-1},0)}}$$
\end{document}