---
title: "Infeasible Primal-Dual Quadratic Programming with Box Constraints"
author: "Jan de Leeuw"
date: "Version 08, May 09, 2017"
output:
html_document:
keep_md: yes
number_sections: yes
toc: yes
pdf_document:
keep_tex: yes
number_sections: yes
toc: yes
toc_depth: 3
fontsize: 12pt
graphics: yes
bibliography: boxqp.bib
abstract: This describes a C version of a infeasible primal-dual algorithm for positive
definite quadratic programming with box constraints, proposed by Voglis and Lagaris.
We also give a straightforward .C() interface for R.
---
```{r function_code, echo = FALSE}
source("boxqp.R")
```
```{r packages, echo = FALSE}
options (digits = 10)
library (captioner)
mprint <- function (x,
d = 6,
w = 8,
f = "") {
print (noquote (formatC (
x,
di = d,
wi = w,
fo = "f",
flag = f
)))
}
```
```{r equation_numbering, echo = FALSE}
figure_nums <- captioner (prefix = "Figure")
theorem_nums <- captioner (prefix = "Theorem")
lemma_nums <- captioner (prefix = "Lemma")
corollary_nums <- captioner (prefix = "Corollary")
```
Note: This is a working paper which will be expanded/updated frequently. All suggestions for improvement are welcome. The directory [gifi.stat.ucla.edu/boxqp](http://gifi.stat.ucla.edu/boxqp) has a pdf version, the complete Rmd file with all code chunks, the R and C code, and the bib file.
#Introduction
The problem we address in this paper is minimization of
\begin{equation}\label{E:f}
f(x)=\frac12 x'Cx+d'x
\end{equation}
over $a\leq x\leq b$ (elementwise). Here $C$ is positive definite, so $f$ is strictly convex. Also we assume without loss of generality that $a**b_i\text{ or }(x_i^{(k)}=b_i\text{ and }\mu_i^{(k)}\geq 0)\}\\
S^{(k)}&=\{i\mid a_i**