From 7ff4a4d46dbd64cd8bc7c07d0c7f11f13779443c Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 10 Jun 2013 19:09:15 +0200
Subject: Moving our two main results to a section preceding the introduction
 of our mechanism

---
 approximation.tex | 599 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 599 insertions(+)

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
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+Even though \EDP{} is NP-hard, designing a mechanism for this problem will
+involve being able to find an approximation $\tilde{L}^*(c)$ of $OPT$
+monotonous with respect to coordinate-wise changes of the cost: if $c$ and $c'$
+are two cost vectors such that $c'=(c_i', c_{-i})$ with $c_i' \leq c_i$, then
+we want $\tilde{L}(c')\geq \tilde{L}(c)$. Furthermore, we seek an approximation
+that can be computed in polynomial time.
 
+This approximation will be obtained by introducing a concave optimization
+problem with a constant approximation ratio to \EDP{}
+(Proposition~\ref{prop:relaxation}).  Using Newton's method, it is then
+possible to solve this concave optimization problem to an arbitrary precision.
+However, this approximation breaks the monotonicity of the approximation.
+Finding a monotone approximate solution to the concave problem will be the
+object of (Section~\ref{sec:monotonicity}).
+
+\subsection{A concave relaxation of \EDP}\label{sec:concave}
+
+Let us introduce a new function $L$:
+\begin{equation}\label{eq:our-relaxation}
+\forall\,\lambda\in[0,1]^n,\quad L(\lambda) \defeq
+\log\det\left(I_d + \sum_{i\in\mathcal{N}} \lambda_i x_i\T{x_i}\right),
+\end{equation}
+
+This function is a relaxation of the value function $V$ defined in
+\eqref{modified} in the following sense: $L(\id_S) = V(S)$ for all
+$S\subseteq\mathcal{N}$, where $\id_S$ denotes the indicator vector of $S$.
+
+The optimization program \eqref{eq:non-strategic} extends naturally to such
+a relaxation. We define:
+\begin{equation}\tag{$P_c$}\label{eq:primal}
+    L^*_c \defeq \max_{\lambda\in[0, 1]^{n}}
+    \left\{L(\lambda) \Big| \sum_{i=1}^{n} \lambda_i c_i
+    \leq B\right\}
+\end{equation}
+
+\begin{proposition}\label{prop:relaxation}
+     $   L^*(c) \leq 2 OPT
+        + 2\max_{i\in\mathcal{N}}V(i)$.
+\end{proposition}
+
+The proof of this proposition follows the \emph{pipage rounding} framework of
+\citeN{pipage}.
+
+This framework uses the \emph{multi-linear} extension $F$ of the submodular
+function $V$. Let $P_\mathcal{N}^\lambda(S)$ be the probability of choosing the
+set $S$ if we select each element $i$ in $\mathcal{N}$ independently with
+probability $\lambda_i$:
+\begin{displaymath}
+    P_\mathcal{N}^\lambda(S) \defeq \prod_{i\in S} \lambda_i
+    \prod_{i\in\mathcal{N}\setminus S}( 1 - \lambda_i).
+\end{displaymath}
+Then, the \emph{multi-linear} extension $F$ is defined by:
+\begin{displaymath}
+    F(\lambda) 
+    \defeq \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\big[V(S)\big]
+    = \sum_{S\subseteq\mathcal{N}} P_\mathcal{N}^\lambda(S) V(S)
+\end{displaymath}
+
+For \EDP{}  the multi-linear extension can be written:
+\begin{equation}\label{eq:multi-linear-logdet}
+    F(\lambda) = \mathbb{E}_{S\sim
+    P_\mathcal{N}^\lambda}\bigg[\log\det \big(I_d + \sum_{i\in S} x_i\T{x_i}\big) \Big].
+\end{equation}
+Note that the relaxation $L$ that we introduced in \eqref{eq:our-relaxation},
+follows naturally from the \emph{multi-linear} relaxation by swapping the
+expectation and the $\log\det$ in \eqref{eq:multi-linear-logdet}:
+\begin{displaymath}
+    L(\lambda) = \log\det\left(\mathbb{E}_{S\sim
+    P_\mathcal{N}^\lambda}\bigg[I_d + \sum_{i\in S} x_i\T{x_i} \bigg]\right).
+\end{displaymath}
+
+The proof proceeds as follows:
+\begin{itemize}
+\item First, we prove that $F$ admits the following rounding property: let
+$\lambda$ be a feasible element of $[0,1]^n$, it is possible to trade one
+fractional component of $\lambda$ for another until one of them becomes
+integral, obtaining a new element $\tilde{\lambda}$ which is both feasible and
+for which $F(\tilde{\lambda})\geq F(\lambda)$. Here, by feasibility of a point
+$\lambda$, we mean that it satisfies the budget constraint $\sum_{i=1}^n
+\lambda_i c_i \leq B$.  This rounding property is referred to in the literature
+as \emph{cross-convexity} (see, \emph{e.g.}, \cite{dughmi}), or
+$\varepsilon$-convexity by \citeN{pipage}. This is stated and proven in
+Lemma~\ref{lemma:rounding} and allows us to bound $F$ in terms of $OPT$.
+\item Next, we prove the central result of bounding $L$  appropriately in terms
+of the multi-linear relaxation $F$ (Lemma \ref{lemma:relaxation-ratio}).
+\item Finally, we conclude the proof of Proposition~\ref{prop:relaxation} by
+combining Lemma~\ref{lemma:rounding} and Lemma~\ref{lemma:relaxation-ratio}.
+\end{itemize}
+
+\begin{lemma}[Rounding]\label{lemma:rounding}
+    For any feasible $\lambda\in[0,1]^{n}$, there exists a feasible
+    $\bar{\lambda}\in[0,1]^{n}$ such that at most one of its components is
+    fractional %, that is, lies in $(0,1)$ and:
+     and $F_{\mathcal{N}}(\lambda)\leq F_{\mathcal{N}}(\bar{\lambda})$.
+\end{lemma}
+\begin{proof}
+    We give a rounding procedure which, given a feasible $\lambda$ with at least
+    two fractional components, returns some feasible $\lambda'$ with one less fractional
+    component such that $F(\lambda) \leq F(\lambda')$.
+
+    Applying this procedure recursively yields the lemma's result.
+    Let us consider such a feasible $\lambda$. Let $i$ and $j$ be two
+    fractional components of $\lambda$ and let us define the following
+    function:
+    \begin{displaymath}
+        F_\lambda(\varepsilon) = F(\lambda_\varepsilon)
+        \quad\textrm{where} \quad
+        \lambda_\varepsilon = \lambda + \varepsilon\left(e_i-\frac{c_i}{c_j}e_j\right)
+    \end{displaymath}
+    It is easy to see that if $\lambda$ is feasible, then:
+    \begin{equation}\label{eq:convex-interval}
+        \forall\varepsilon\in\Big[\max\Big(-\lambda_i,(\lambda_j-1)\frac{c_j}{c_i}\Big), \min\Big(1-\lambda_i, \lambda_j
+        \frac{c_j}{c_i}\Big)\Big],\;
+            \lambda_\varepsilon\;\;\textrm{is feasible}
+    \end{equation}
+    Furthermore, the function $F_\lambda$ is convex; indeed:
+    \begin{align*}
+        F_\lambda(\varepsilon)
+        & = \mathbb{E}_{S'\sim P_{\mathcal{N}\setminus\{i,j\}}^\lambda(S')}\Big[
+        (\lambda_i+\varepsilon)\Big(\lambda_j-\varepsilon\frac{c_i}{c_j}\Big)V(S'\cup\{i,j\})\\
+        & + (\lambda_i+\varepsilon)\Big(1-\lambda_j+\varepsilon\frac{c_i}{c_j}\Big)V(S'\cup\{i\})
+         + (1-\lambda_i-\varepsilon)\Big(\lambda_j-\varepsilon\frac{c_i}{c_j}\Big)V(S'\cup\{j\})\\
+         & + (1-\lambda_i-\varepsilon)\Big(1-\lambda_j+\varepsilon\frac{c_i}{c_j}\Big)V(S')\Big]
+    \end{align*}
+    Thus, $F_\lambda$ is a degree 2 polynomial whose dominant coefficient is:
+    \begin{displaymath}
+        \frac{c_i}{c_j}\mathbb{E}_{S'\sim
+        P_{\mathcal{N}\setminus\{i,j\}}^\lambda(S')}\Big[
+            V(S'\cup\{i\})+V(S'\cup\{i\})\\
+        -V(S'\cup\{i,j\})-V(S')\Big]
+    \end{displaymath}
+    which is positive by submodularity of $V$. Hence, the maximum of
+    $F_\lambda$ over the interval given in \eqref{eq:convex-interval} is
+    attained at one of its limit, at which either the $i$-th or $j$-th component of
+    $\lambda_\varepsilon$ becomes integral.
+\end{proof}
+
+\begin{lemma}\label{lemma:relaxation-ratio}
+ %   The following inequality holds:
+For all $\lambda\in[0,1]^{n},$
+    %\begin{displaymath}
+     $           \frac{1}{2}
+        \,L(\lambda)\leq
+        F(\lambda)\leq L(\lambda)$.
+    %\end{displaymath}
+\end{lemma}
+\begin{proof}
+    The bound $F_{\mathcal{N}}(\lambda)\leq L_{\mathcal{N}(\lambda)}$ follows by the concavity of the $\log\det$ function.
+    To show the lower bound, 
+    we first prove that $\frac{1}{2}$ is a lower bound of the ratio $\partial_i
+    F(\lambda)/\partial_i L(\lambda)$, where
+    $\partial_i\, \cdot$ denotes the partial derivative with respect to the
+    $i$-th variable.  
+
+   Let us start by computing the derivatives of $F$ and
+    $L$ with respect to the $i$-th component.
+    Observe that
+    \begin{displaymath}
+        \partial_i P_\mathcal{N}^\lambda(S) = \left\{
+            \begin{aligned}
+                & P_{\mathcal{N}\setminus\{i\}}^\lambda(S\setminus\{i\})\;\textrm{if}\;
+                i\in S, \\
+                & - P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\;\textrm{if}\;
+                i\in \mathcal{N}\setminus S. \\
+            \end{aligned}\right.
+    \end{displaymath}
+    Hence,
+    \begin{displaymath}
+        \partial_i F(\lambda) =
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S\setminus\{i\})V(S)
+        - \sum_{\substack{S\subseteq\mathcal{N}\\ i\in \mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)V(S).
+    \end{displaymath}
+    Now, using that every $S$ such that $i\in S$ can be uniquely written as
+    $S'\cup\{i\}$, we can write:
+    \begin{displaymath}
+        \partial_i F(\lambda) =
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\big(V(S\cup\{i\})
+        - V(S)\big).
+    \end{displaymath}
+    The marginal contribution of $i$ to
+    $S$ can be written as 
+\begin{align*}
+V(S\cup \{i\}) - V(S)& =   \frac{1}{2}\log\det(I_d 
+    + \T{X_S}X_S + x_i\T{x_i})
+    - \frac{1}{2}\log\det(I_d + \T{X_S}X_S)\\
+    &  = \frac{1}{2}\log\det(I_d + x_i\T{x_i}(I_d +
+\T{X_S}X_S)^{-1})
+ = \frac{1}{2}\log(1 + \T{x_i}A(S)^{-1}x_i)
+\end{align*}
+where $A(S) \defeq I_d+ \T{X_S}X_S$, and the last equality follows from the
+Sylvester's determinant identity~\cite{sylvester}.
+%  $  V(S\cup\{i\}) - V(S) = \frac{1}{2}\log\left(1 +  \T{x_i} A(S)^{-1}x_i\right)$.
+Using this,
+    \begin{displaymath}
+        \partial_i F(\lambda) = \frac{1}{2}
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)
+        \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big)
+    \end{displaymath}
+     The computation of the derivative of $L$ uses standard matrix
+    calculus: writing $\tilde{A}(\lambda) = I_d+\sum_{i\in
+    \mathcal{N}}\lambda_ix_i\T{x_i}$,
+    \begin{displaymath}
+        \det \tilde{A}(\lambda + h\cdot e_i) = \det\big(\tilde{A}(\lambda)
+        + hx_i\T{x_i}\big)
+         =\det \tilde{A}(\lambda)\big(1+
+        h\T{x_i}\tilde{A}(\lambda)^{-1}x_i\big).
+    \end{displaymath}
+    Hence,
+    \begin{displaymath}
+       \log\det\tilde{A}(\lambda + h\cdot e_i)= \log\det\tilde{A}(\lambda)
+        + h\T{x_i}\tilde{A}(\lambda)^{-1}x_i + o(h),
+    \end{displaymath}
+    which implies
+    \begin{displaymath}
+        \partial_i L(\lambda)
+        =\frac{1}{2} \T{x_i}\tilde{A}(\lambda)^{-1}x_i.
+    \end{displaymath}
+
+For two symmetric matrices $A$ and $B$, we write $A\succ B$ ($A\succeq B$) if
+$A-B$ is positive definite (positive semi-definite).  This order allows us to
+define the notion of a \emph{decreasing} as well as  \emph{convex} matrix
+function, similarly to their real counterparts. With this definition, matrix
+inversion is decreasing and convex over symmetric positive definite
+matrices (see Example 3.48 p. 110 in \cite{boyd2004convex}).
+In particular,
+\begin{gather*}
+        \forall S\subseteq\mathcal{N},\quad A(S)^{-1} \succeq A(S\cup\{i\})^{-1}
+\end{gather*}
+as $A(S)\preceq A(S\cup\{i\})$. Observe that
+ %   \begin{gather*}
+ %       \forall 
+$P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\geq P_{\mathcal{N}\setminus\{i\}}^\lambda(S\cup\{i\})$ for all $S\subseteq\mathcal{N}\setminus\{i\}$, and
+       % ,\\
+        $P_{\mathcal{N}\setminus\{i\}}^\lambda(S) \geq P_\mathcal{N}^\lambda(S),$ for all $S\subseteq\mathcal{N}$.
+ %\end{gather*}
+    Hence,
+    \begin{align*}
+        \partial_i F(\lambda) 
+      %  & = \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}} P_\mathcal{N}^\lambda(S) \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big)\\
+        & \geq \frac{1}{4}
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)
+        \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big)\\
+        &\hspace{-3.5em}+\frac{1}{4}
+        \sum_{\substack{S\subseteq\mathcal{N}\\ i\in\mathcal{N}\setminus S}}
+        P_{\mathcal{N}\setminus\{i\}}^\lambda(S\cup\{i\})
+        \log\Big(1 + \T{x_i}A(S\cup\{i\})^{-1}x_i\Big)\\
+        &\geq \frac{1}{4}
+        \sum_{S\subseteq\mathcal{N}}
+        P_\mathcal{N}^\lambda(S)
+        \log\Big(1 + \T{x_i}A(S)^{-1}x_i\Big).
+    \end{align*}
+    Using that $A(S)\succeq I_d$ we get that $\T{x_i}A(S)^{-1}x_i \leq
+    \norm{x_i}_2^2 \leq 1$. Moreover, $\log(1+x)\geq x$ for all $x\leq 1$.
+    Hence,
+    \begin{displaymath}
+        \partial_i F(\lambda) \geq
+        \frac{1}{4}
+        \T{x_i}\bigg(\sum_{S\subseteq\mathcal{N}}P_\mathcal{N}^\lambda(S)A(S)^{-1}\bigg)x_i.
+    \end{displaymath}
+    Finally, using that the inverse is a matrix convex function over symmetric
+    positive definite matrices:
+    \begin{displaymath}
+        \partial_i F(\lambda) \geq
+        \frac{1}{4}
+        \T{x_i}\bigg(\sum_{S\subseteq\mathcal{N}}P_\mathcal{N}^\lambda(S)A(S)\bigg)^{-1}x_i
+        = \frac{1}{4}\T{x_i}\tilde{A}(\lambda)^{-1}x_i
+         = \frac{1}{2}
+     \partial_i L(\lambda).
+    \end{displaymath}
+
+Having bound the ratio between the partial derivatives, we now bound the ratio $F(\lambda)/L(\lambda)$ from below. Consider the following cases.
+    First, if the minimum of the ratio
+    $F(\lambda)/L(\lambda)$ is attained at a point interior to the hypercube, then it is
+    a critical point, \emph{i.e.}, $\partial_i \big(F(\lambda)/L(\lambda)\big)=0$ for all $i\in \mathcal{N}$; hence, at such a critical point:
+    \begin{equation}\label{eq:lhopital}
+        \frac{F(\lambda)}{L(\lambda)}
+        = \frac{\partial_i F(\lambda)}{\partial_i
+        L(\lambda)} \geq \frac{1}{2}.
+    \end{equation}
+    Second, if the minimum is attained as 
+    $\lambda$ converges to zero in, \emph{e.g.}, the $l_2$ norm, by the Taylor approximation, one can write:
+    \begin{displaymath}
+        \frac{F(\lambda)}{L(\lambda)}
+        \sim_{\lambda\rightarrow 0}
+        \frac{\sum_{i\in \mathcal{N}}\lambda_i\partial_i F(0)}
+        {\sum_{i\in\mathcal{N}}\lambda_i\partial_i L(0)}
+        \geq \frac{1}{2},
+    \end{displaymath}
+    \emph{i.e.}, the ratio $\frac{F(\lambda)}{L(\lambda)}$ is necessarily bounded from below by 1/2 for small enough $\lambda$. 
+       Finally, if the minimum is attained on a face of the hypercube $[0,1]^n$ (a face is
+    defined as a subset of the hypercube where one of the variable is fixed to
+    0 or 1), without loss of generality, we can assume that the minimum is
+    attained on the face where the $n$-th variable has been fixed
+    to 0 or 1. Then, either the minimum is attained at a point interior to the
+    face or on a boundary of the face. In the first sub-case, relation
+    \eqref{eq:lhopital} still characterizes the minimum for $i< n$.
+    In the second sub-case, by repeating the argument again by induction, we see
+    that all is left to do is to show that the bound holds for the vertices of
+    the cube (the faces of dimension 1).  The vertices are exactly the binary
+    points, for which we know that both relaxations are equal to the value
+    function $V$. Hence, the ratio is equal to 1 on the vertices.
+\end{proof}
+
+To conclude the proof of Proposition~\ref{prop:relaxation}, let us consider
+a feasible point $\lambda^*\in[0,1]^{n}$ such that $L(\lambda^*) = L^*_c$. By
+applying Lemma~\ref{lemma:relaxation-ratio} and Lemma~\ref{lemma:rounding} we
+get a feasible point $\bar{\lambda}$ with at most one fractional component such
+that
+\begin{equation}\label{eq:e1}
+    L(\lambda^*) \leq 2 F(\bar{\lambda}).
+\end{equation}
+    Let $\lambda_i$ denote the fractional component of $\bar{\lambda}$ and $S$
+    denote the set whose indicator vector is $\bar{\lambda} - \lambda_i e_i$.
+    By definition of the multi-linear extension $F$:
+    \begin{displaymath}
+        F(\bar{\lambda}) = (1-\lambda_i)V(S) +\lambda_i V(S\cup\{i\}).
+    \end{displaymath}
+    By submodularity of $V$, $V(S\cup\{i\})\leq V(S) + V(\{i\})$. Hence,
+    \begin{displaymath}
+        F(\bar{\lambda}) \leq V(S) + V(i).
+    \end{displaymath}
+    Note that since $\bar{\lambda}$ is feasible, $S$ is also feasible and
+    $V(S)\leq OPT$. Hence,
+    \begin{equation}\label{eq:e2}
+        F(\bar{\lambda}) \leq  OPT + \max_{i\in\mathcal{N}} V(i).
+    \end{equation}
+    Together, \eqref{eq:e1} and \eqref{eq:e2} imply the lemma. \hspace*{\stretch{1}}\qed
+
+\subsection{A monotonous Newton's estimator}\label{sec:monotonicity}
+
+\textbf{TODO} Explain that we only get approximate monotonicity, but even
+that is not immediate since the variation induced by a change of cost on
+coordinate $i$ depends on the allocation at this coordinate which can be
+arbitrarily small.
+
+For the ease of presentation, we normalize the costs by dividing them by the
+budget $B$ so that the budget constraint in \eqref{eq:primal} now reads
+$\T{c}\lambda\leq 1$.
+
+\begin{proposition}\label{prop:monotonicity}
+    Let $\delta\in(0,1]$. For any $\varepsilon\in(0,1]$, there exists
+    an algorithm which computes an approximate solution $\tilde{L}^*_c$ to
+    \eqref{eq:primal} such that:
+    \begin{enumerate}
+        \item $|\tilde{L}^*_c - L^*_c| \leq \varepsilon$
+        \item for all $c' = (c_i', c_{-i})$ with $c_i'\leq c_i-\delta$, $\tilde{L}^*_c \leq \tilde{L}^*_{c'}$
+        \item the routine's running time is $O\big(poly(n, d, \log\log\frac{1}{b\varepsilon\delta})\big)$
+    \end{enumerate}
+\end{proposition}
+
+We consider a perturbation of \eqref{eq:primal} by introducing:
+\begin{equation}\tag{$P_{c, \alpha}$}\label{eq:perturbed-primal}
+    L^*_c(\alpha) \defeq \max_{\lambda\in[\alpha, 1]^{n}}
+    \left\{L(\lambda) \Big| \sum_{i=1}^{n} \lambda_i c_i
+    \leq B\right\}
+\end{equation}
+Note that we have $L^*_c = L^*_c(0)$.  We will also assume that
+$\alpha<\frac{1}{n}$ so that \eqref{eq:perturbed-primal} has at least one
+feasible point: $(\frac{1}{n},\ldots,\frac{1}{n})$.
+
+Having introduced this perturbed problem, we show that its optimal value is
+close to the optimal value of \eqref{eq:primal} (Lemma~\ref{lemma:proximity})
+while being well-behaved with respect to changes of the cost
+(Lemma~\ref{lemma:monotonicity}). These lemmas together imply
+Proposition~\ref{prop:monotonicity}.
+
+
+\begin{lemma}\label{lemma:derivative-bounds}
+    Let $\partial_i L(\lambda)$ denote the $i$-th derivative of $L$, for $i\in\{1,\ldots, n\}$, then:
+    \begin{displaymath}
+        \forall\lambda\in[0, 1]^n,\;\frac{b}{2^n} \leq \partial_i L(\lambda) \leq 1
+    \end{displaymath}
+\end{lemma}
+
+\begin{proof}
+    Let us define:
+    \begin{displaymath}
+        S(\lambda)\defeq I_d + \sum_{i=1}^n \lambda_i x_i\T{x_i}
+        \quad\mathrm{and}\quad
+        S_k \defeq I_d + \sum_{i=1}^k x_i\T{x_i}
+    \end{displaymath}
+
+    We have $\partial_i L(\lambda) = \T{x_i}S(\lambda)^{-1}x_i$. Since
+    $S(\lambda)\geq I_d$, $\partial_i L(\lambda)\leq \T{x_i}x_i \leq 1$, which
+    is the right-hand side  of the lemma.
+
+    For the left-hand side, note that $S(\lambda) \leq S_n$. Hence
+    $\partial_iL(\lambda)\geq \T{x_i}S_n^{-1}x_i$.
+
+    Using the Sherman-Morrison formula, for all $k\geq 1$:
+    \begin{displaymath}
+        \T{x_i}S_k^{-1} x_i = \T{x_i}S_{k-1}^{-1}x_i 
+        - \frac{(\T{x_i}S_{k-1}^{-1}x_k)^2}{1+\T{x_k}S_{k-1}^{-1}x_k}
+    \end{displaymath}
+
+    By the Cauchy-Schwarz inequality:
+    \begin{displaymath}
+        (\T{x_i}S_{k-1}^{-1}x_k)^2 \leq \T{x_i}S_{k-1}^{-1}x_i\;\T{x_k}S_{k-1}^{-1}x_k
+    \end{displaymath}
+
+    Hence:
+    \begin{displaymath}
+        \T{x_i}S_k^{-1} x_i \geq \T{x_i}S_{k-1}^{-1}x_i 
+        - \T{x_i}S_{k-1}^{-1}x_i\frac{\T{x_k}S_{k-1}^{-1}x_k}{1+\T{x_k}S_{k-1}^{-1}x_k}
+    \end{displaymath}
+
+    But $\T{x_k}S_{k-1}^{-1}x_k\leq 1$ and $\frac{a}{1+a}\leq \frac{1}{2}$ if
+    $0\leq a\leq 1$, so:
+    \begin{displaymath}
+        \T{x_i}S_{k}^{-1}x_i \geq \T{x_i}S_{k-1}^{-1}x_i
+        - \frac{1}{2}\T{x_i}S_{k-1}^{-1}x_i\geq \frac{\T{x_i}S_{k-1}^{-1}x_i}{2}
+    \end{displaymath}
+
+    By induction:
+    \begin{displaymath}
+        \T{x_i}S_n^{-1} x_i \geq \frac{\T{x_i}x_i}{2^n}
+    \end{displaymath}
+    
+    Using that $\T{x_i}{x_i}\geq b$ concludes the proof of the left-hand side
+    of the lemma's inequality.
+\end{proof}
+
+Let us introduce the lagrangian of problem, \eqref{eq:perturbed-primal}:
+
+\begin{displaymath}
+    \mathcal{L}_{c, \alpha}(\lambda, \mu, \nu, \xi) \defeq L(\lambda) 
+    + \T{\mu}(\lambda-\alpha\mathbf{1}) + \T{\nu}(\mathbf{1}-\lambda) + \xi(1-\T{c}\lambda)
+\end{displaymath}
+so that:
+\begin{displaymath}
+    L^*_c(\alpha) = \min_{\mu, \nu, \xi\geq 0}\max_\lambda \mathcal{L}_{c, \alpha}(\lambda, \mu, \nu, \xi)
+\end{displaymath}
+Similarly, we define $\mathcal{L}_{c}\defeq\mathcal{L}_{c, 0}$ the lagrangian of \eqref{eq:primal}.
+
+Let $\lambda^*$ be primal optimal for \eqref{eq:perturbed-primal}, and $(\mu^*,
+\nu^*, \xi^*)$ be dual optimal for the same problem. In addition to primal and
+dual feasibility, the KKT conditions give $\forall i\in\{1, \ldots, n\}$:
+\begin{gather*}
+    \partial_i L(\lambda^*) + \mu_i^* - \nu_i^* - \xi^* c_i = 0\\
+    \mu_i^*(\lambda_i^* - \alpha) = 0\\
+    \nu_i^*(1 - \lambda_i^*) = 0
+\end{gather*}
+
+\begin{lemma}\label{lemma:proximity}
+We have:
+\begin{displaymath}
+    L^*_c - \alpha n^2\leq L^*_c(\alpha) \leq L^*_c
+\end{displaymath}
+In particular, $|L^*_c - L^*_c(\alpha)| \leq \alpha n^2$.
+\end{lemma}
+
+\begin{proof}
+    $\alpha\mapsto L^*_c(\alpha)$ is a decreasing function as it is the
+    maximum value of the $L$ function over a set-decreasing domain, which gives
+    the rightmost inequality.
+
+    Let $\mu^*, \nu^*, \xi^*$ be dual optimal for $(P_{c, \alpha})$, that is:
+    \begin{displaymath}
+        L^*_{c}(\alpha) = \max_\lambda \mathcal{L}_{c, \alpha}(\lambda, \mu^*, \nu^*, \xi^*)
+    \end{displaymath}
+
+    Note that $\mathcal{L}_{c, \alpha}(\lambda, \mu^*, \nu^*, \xi^*)
+    = \mathcal{L}_{c}(\lambda, \mu^*, \nu^*, \xi^*)
+    - \alpha\T{\mathbf{1}}\mu^*$, and that for any $\lambda$ feasible for
+    problem \eqref{eq:primal}, $\mathcal{L}_{c}(\lambda, \mu^*, \nu^*, \xi^*)
+    \geq L(\lambda)$. Hence,
+    \begin{displaymath}
+        L^*_{c}(\alpha) \geq L(\lambda) - \alpha\T{\mathbf{1}}\mu^*
+    \end{displaymath}
+    for any $\lambda$ feasible for \eqref{eq:primal}. In particular, for $\lambda$ primal optimal for $\eqref{eq:primal}$:
+    \begin{equation}\label{eq:local-1}
+        L^*_{c}(\alpha) \geq L^*_c - \alpha\T{\mathbf{1}}\mu^*
+    \end{equation}
+
+    Let us denote by the $M$ the support of $\mu^*$, that is $M\defeq
+    \{i|\mu_i^* > 0\}$, and by $\lambda^*$ a primal optimal point for
+    $\eqref{eq:perturbed-primal}$.  From the KKT conditions we see that:
+    \begin{displaymath}
+        M \subseteq \{i|\lambda_i^* = \alpha\}
+    \end{displaymath}
+
+
+    Let us first assume that $|M| = 0$, then $\T{\mathbf{1}}\mu^*=0$ and the lemma follows.
+
+    We will now assume that $|M|\geq 1$. In this case $\T{c}\lambda^*
+    = 1$, otherwise we could increase the coordinates of $\lambda^*$ in $M$,
+    which would increase the value of the objective function and contradict the
+    optimality of $\lambda^*$. Note also, that $|M|\leq n-1$, otherwise, since
+    $\alpha< \frac{1}{n}$, we would have $\T{c}\lambda^*\ < 1$, which again
+    contradicts the optimality of $\lambda^*$. Let us write:
+    \begin{displaymath}
+        1 = \T{c}\lambda^* = \alpha\sum_{i\in M}c_i + \sum_{i\in \bar{M}}\lambda_i^*c_i
+        \leq \alpha |M| + (n-|M|)\max_{i\in \bar{M}} c_i
+    \end{displaymath}
+    That is:
+    \begin{equation}\label{local-2}
+        \max_{i\in\bar{M}} c_i \geq \frac{1 - |M|\alpha}{n-|M|}> \frac{1}{n}
+    \end{equation}
+    where the last inequality uses again that $\alpha<\frac{1}{n}$. From the
+    KKT conditions, we see that for $i\in M$, $\nu_i^* = 0$ and:
+    \begin{equation}\label{local-3}
+        \mu_i^* = \xi^*c_i - \partial_i L(\lambda^*)\leq \xi^*c_i\leq \xi^*
+    \end{equation}
+    since $\partial_i L(\lambda^*)\geq 0$ and $c_i\leq 1$.
+
+    Furthermore, using the KKT conditions again, we have that:
+    \begin{equation}\label{local-4}
+        \xi^* \leq \inf_{i\in \bar{M}}\frac{\partial_i L(\lambda^*)}{c_i}\leq \inf_{i\in \bar{M}} \frac{1}{c_i}
+        = \frac{1}{\max_{i\in\bar{M}} c_i}
+    \end{equation}
+    where the last inequality uses Lemma~\ref{lemma:derivative-bounds}.
+
+    Combining \eqref{local-2}, \eqref{local-3} and \eqref{local-4}, we get that:
+    \begin{displaymath}
+        \sum_{i\in M}\mu_i^* \leq |M|\xi^* \leq n\xi^*\leq \frac{n}{\max_{i\in\bar{M}} c_i} \leq n^2
+    \end{displaymath}
+    
+    This implies that:
+    \begin{displaymath}
+        \T{\mathbf{1}}\mu^* = \sum_{i=1}^n \mu^*_i = \sum_{i\in M}\mu_i^*\leq n^2
+    \end{displaymath}
+    which in addition to \eqref{eq:local-1} proves the lemma.
+\end{proof} 
+
+\begin{lemma}\label{lemma:monotonicity}
+    If $c'$ = $(c_i', c_{-i})$, with $c_i'\leq c_i - \delta$, we have:
+    \begin{displaymath}
+        L^*_{c'}(\alpha) \geq L^*_c(\alpha) + \frac{\alpha\delta b}{2^n}
+    \end{displaymath}
+\end{lemma}
+
+\begin{proof}
+    Let $\mu^*, \nu^*, \xi^*$ be dual optimal for $(P_{c', \alpha})$. Noting that:
+    \begin{displaymath} 
+    \mathcal{L}_{c', \alpha}(\lambda, \mu^*, \nu^*, \xi^*) \geq
+    \mathcal{L}_{c, \alpha}(\lambda, \mu^*, \nu^*, \xi^*) + \lambda_i\xi^*\delta,
+    \end{displaymath}
+    we get similarly to Lemma~\ref{lemma:proximity}:
+    \begin{displaymath}
+        L^*_{c'}(\alpha) \geq L(\lambda) + \lambda_i\xi^*\delta
+    \end{displaymath}
+    for any $\lambda$ feasible for \eqref{eq:perturbed-primal}. In particular, for $\lambda^*$ primal optimal for \eqref{eq:perturbed-primal}:
+    \begin{displaymath}
+        L^*_{c'}(\alpha) \geq L^*_{c}(\alpha) + \alpha\xi^*\delta
+    \end{displaymath}
+    since $\lambda_i^*\geq \alpha$.
+
+    Using the KKT conditions for $(P_{c', \alpha})$, we can write:
+    \begin{displaymath}
+        \xi^* = \inf_{i:\lambda^{'*}_i>\alpha} \frac{\T{x_i}S(\lambda^{'*})^{-1}x_i}{c_i'}
+    \end{displaymath}
+    with $\lambda^{'*}$ optimal for $(P_{c', \alpha})$. Since $c_i'\leq 1$, using Lemma~\ref{lemma:derivative-bounds}, we get that $\xi^*\geq \frac{b}{2^n}$, which concludes the proof.
+\end{proof}
+
+
+\subsubsection*{End of the proof of Proposition~\ref{prop:monotonicity}}
+
+Let $\varepsilon$ in $(0, 1]$. The routine works as follows: set $\alpha\defeq
+\varepsilon(\delta + n^2)^{-1}$ and return an approximation $\tilde{L}^*_c$ of
+$L^*_c(\alpha)$ with an accuracy $\frac{1}{2^{n+1}}\alpha\delta b$ computed by
+a standard convex optimization algorithm. Note that this choice of $\alpha$
+implies $\alpha<\frac{1}{n}$ as required.
+
+\begin{enumerate}
+    \item using Lemma~\ref{lemma:proximity}:
+\begin{displaymath}
+        |\tilde{L}^*_c - L^*_c| \leq |\tilde{L}^*_c - L^*_c(\alpha)| + |L^*_c(\alpha) - L^*_c|
+        \leq \alpha\delta + \alpha n^2 = \varepsilon
+\end{displaymath}
+
+\item let $c' = (c_i', c_{-i})$ with $c_i'\leq c_i-\delta$, then:
+\begin{displaymath}
+    \tilde{L}^*_{c'} \geq L^*_{c'} - \frac{\alpha\delta b}{2^{n+1}} 
+                     \geq L^*_c + \frac{\alpha\delta b}{2^{n+1}}
+    \geq \tilde{L}^*_c
+\end{displaymath}
+where the first and inequality come from the accuracy of the approximation, and
+the inner inequality follows from Lemma~\ref{lemma:monotonicity}.
+
+\item the accuracy of the approximation $\tilde{L}^*_c$ is:
+\begin{displaymath}
+    A\defeq\frac{\varepsilon\delta b}{2^{n+1}(\delta + n^2)}
+\end{displaymath}
+
+The function $L$ is well-known to be concave and even self-concordant (see
+\emph{e.g.}, \cite{boyd2004convex}).  In this case, the analysis of Newton's
+method for self-concordant functions in \cite{boyd2004convex}, shows that
+finding the maximum of $L$ to any precision $A$ can be done in
+$O(\log\log A^{-1})$ iterations. Note that:
+\begin{displaymath}
+    \log\log A^{-1} = O\bigg(\log\log\frac{1}{\varepsilon\delta b} + \log n\bigg)
+\end{displaymath}
+Furthermore, each iteration of Newton's method can be done in time $O\big(poly(n,
+d)\big)$.\qed
+\end{enumerate}
-- 
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