From 62c98a8cc5c9f3feb18ccc4ccc2c5d11d850c020 Mon Sep 17 00:00:00 2001 From: Stratis Ioannidis Date: Sat, 6 Jul 2013 13:47:50 -0700 Subject: linear constraints --- approximation.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'approximation.tex') diff --git a/approximation.tex b/approximation.tex index c306a75..aed79a0 100644 --- a/approximation.tex +++ b/approximation.tex @@ -112,7 +112,7 @@ The proof of this proposition can be found in Appendix~\ref{proofofproprelaxatio The $\log\det$ objective function of \eqref{eq:primal} is strictly concave and \emph{self-concordant} \cite{boyd2004convex}. The maximization of a concave, self-concordant function subject to a set of linear constraints can be performed through the \emph{barrier method} (see, \emph{e.g.}, \cite{boyd2004convex} Section 11.5.5 for general self-concordant optimization as well as \cite{vandenberghe1998determinant} for a detailed treatment of the $\log\det$ objective). The performance of the barrier method is summarized in our case by the following lemma: \begin{lemma}[\citeN{boyd2004convex}]\label{lemma:barrier} 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 $O\left(poly(n,d,\log\log\varepsilon^{-1})\right)$. +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 for 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. -- cgit v1.2.3-70-g09d2