small

1 files changed, 3 insertions, 4 deletions

diff --git a/approximation.tex b/approximation.tex
index 1b94615..546961f 100755
--- a/approximation.tex
+++ b/approximation.tex
@@ -123,13 +123,12 @@ Nevertheless, we prove that it is possible to use the barrier method to construc
     for all $i\in \mathcal{N},x\in\reals^n, \mu\geq\delta,$
 %\end{equation}
 where $e_i$ is the $i$-th canonical basis vector of $\reals^n$.
-In other words, $f$ is $\delta$-decreasing if increasing any coordinate by $\delta$ or more at input $x$ ensures that the output will be at most $f(x)$. In particular, the following proposition holds:
+In other words, $f$ is $\delta$-decreasing if increasing any coordinate by $\delta$ or more at input $x$ ensures that the output will be at most $f(x)$. The following holds:
 
 \begin{proposition}\label{prop:monotonicity}
     For any $\delta\in(0,1]$ and any $\varepsilon\in(0,1]$,
-    Algorithm~\ref{alg:monotone} computes a $\delta$-decreasing,
+    there exists an algorithm computes a $\delta$-decreasing,
     $\varepsilon$-accurate approximation of $L^*_c$. The running time of the
     algorithm is $O\big(poly(n, d, \log\log\frac{B}{b\varepsilon\delta})\big)$.
 \end{proposition}
-
-The description of Algorithm~\ref{alg:monotone} as well as the proof of the proposition can be found in Appendix~\ref{proofofproprelaxation}.
+The proof and the  algorithm (Algorithm~\ref{alg:monotone}) are in Appendix~\ref{proofofproprelaxation}.