deletions | additions       diff --git a/paragraph_RTM_runs_in_polynomial__.tex b/paragraph_RTM_runs_in_polynomial__.tex  index 6f4d6ba..977990e 100644  --- a/paragraph_RTM_runs_in_polynomial__.tex  +++ b/paragraph_RTM_runs_in_polynomial__.tex 
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This is proved by showing   \begin{enumerate}  \item in $R_{II}$ and $R_{III}$, each step reduces the objective values by at least a constant (Proposition 3.7), and  \item in $R_{I}$, the algorithm first takes \textit{constrained steps} and then  followed by \textit{all unconstrained only \textit{unconstrained  steps} \item objective is reduced by at least a fixed amount for each constrained step in $R_{I}$, and   \item the distance to the local minimum drops quadratically for unconstrained steps in $R_I$.   \end{enumerate}