From b1afe9b3e8d876f4ddb7e8364c115ea41379457f Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Sun, 7 Jul 2013 22:13:12 -0700
Subject: minor edits

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

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
index b1d3453..5e5ee11 100644
--- a/approximation.tex
+++ b/approximation.tex
@@ -115,7 +115,7 @@ 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 $O\left(poly(n,d,\log\log\varepsilon^{-1})\right)$. The same guarantees apply when 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.
+ 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.
 
 Nevertheless, we prove that it is possible to use the barrier method to construct an approximation of $L^*_{c}$ that is ``almost'' monotone. More specifically, given $\delta>0$,  we say that $f:\reals^n\to\reals$ is
 \emph{$\delta$-decreasing}  if
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