From cb490cb5dc8a6f76e87b6d130c2fc6b9150c9936 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 23 Dec 2013 16:26:21 +0100
Subject: Ultimate changes before submission

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

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
index 2162f14..3a4376b 100755
--- a/approximation.tex
+++ b/approximation.tex
@@ -136,7 +136,7 @@ Note that $\dom_c=\dom_{c,0}.$ Consider the following perturbed problem:
 \end{split}
 \end{align}
 
-The restricted set $\dom_{c,\alpha}$ ensures that the partial derivatives of the optimal solution to $P_{c,\alpha}$ with respect to the costs are bounded from below. This implies that an approximate solution to $P_{c,\alpha}$ given by the barrier method is $\delta$-decreasing with respect to the costs. On the other hand, by taking $\alpha$ small enough, we ensure that the approximate solution to $P_{c,\alpha}$ is still an $\epsilon$-accurate approximation of $L_c^*$. This methodology is summarized in the following proposition whose proof can be found in \cite{arxiv}.
+Restricting the feasible set to $\dom_{c,\alpha}$ ensures that the gradient of the optimal solution with respect to $c$ is bounded from below. This implies that an approximate solution to $P_{c,\alpha}$ given by the barrier method is $\delta$-decreasing with respect to the costs. On the other hand, by taking $\alpha$ small enough, we ensure that the approximate solution to $P_{c,\alpha}$ is still an $\epsilon$-accurate approximation of $L_c^*$. This methodology is summarized in the following proposition, whose proof can be found in \cite{arxiv}.
 
 \begin{proposition}\label{prop:monotonicity}
     For any $\delta\in(0,1]$ and any $\varepsilon\in(0,1]$, using the barrier
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