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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 |