From 62c98a8cc5c9f3feb18ccc4ccc2c5d11d850c020 Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Sat, 6 Jul 2013 13:47:50 -0700
Subject: linear constraints

---
 approximation.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
index c306a75..aed79a0 100644
--- a/approximation.tex
+++ b/approximation.tex
@@ -112,7 +112,7 @@ The proof of this proposition can be found in Appendix~\ref{proofofproprelaxatio
  The $\log\det$ objective function of \eqref{eq:primal} is strictly concave and \emph{self-concordant} \cite{boyd2004convex}. The maximization of a concave, self-concordant function subject to a set of linear constraints can be performed through the \emph{barrier method} (see, \emph{e.g.}, \cite{boyd2004convex} Section 11.5.5 for  general self-concordant optimization as well as \cite{vandenberghe1998determinant} for a detailed treatment of the $\log\det$ objective). The performance of the barrier method is summarized in our case by the following lemma:
 \begin{lemma}[\citeN{boyd2004convex}]\label{lemma:barrier}
 For any $\varepsilon>0$, the barrier method computes  an 
-approximation $\hat{L}^*_c$ that is $\varepsilon$-accurate, \emph{i.e.}, it satisfies $|\hat L^*_c- L^*_c|\leq \varepsilon$, in time $O\left(poly(n,d,\log\log\varepsilon^{-1})\right)$.
+approximation $\hat{L}^*_c$ that is $\varepsilon$-accurate, \emph{i.e.}, it satisfies $|\hat L^*_c- L^*_c|\leq \varepsilon$, in time $O\left(poly(n,d,\log\log\varepsilon^{-1})\right)$. The same guarantees apply for maximizing $L$ subject to an arbitrary set of $O(n)$ linear constraints.
 \end{lemma}
 
  Clearly, the optimal value $L^*_c$ of \eqref{eq:primal} is monotone in $c$. Formally, for any two $c,c'\in \reals_+^n$ s.t.~$c\leq c'$ coordinate-wise, $\dom_{c'}\subseteq \dom_c$ and thus $L^*_c\geq L^*_{c'}$. Hence, the map $c\mapsto L^*_c$ is non-increasing. Unfortunately, the same is not true for the output $\hat{L}_c^*$ of the barrier method: there is no guarantee that the $\epsilon$-accurate approximation $\hat{L}^*_c$ exhibits any kind of monotonicity.
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