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poster.tex

@@ -56,14 +56,14 @@ We consider the incentive allocation problem with additional constraints.
 }
 
 This problem is NP-hard. Consider its LP relaxation.
-\begin{equation}\label{LP}
+\begin{equation*}\label{LP}
 \begin{aligned}
 \tau(B)=\max_x&\; & v\cdot x& & & \\
 s.t.&\;  & c \cdot x &\leq B & &\\
 & & x_{E_i}&\in \conv(\mathcal{F}_i)  & &\;\forall i\in [n]\\
 & & x&\in [0,1]^m & &
 \end{aligned}
-\end{equation}
+\end{equation*}
 \textbf{Output}: The entire curve $\tau(B)$ for $B\in [0,\infty)$.
 
 We consider 3 cases of additional constraints $x_{E_i}\in \mathcal{F}_i$ :
@@ -106,12 +106,12 @@ The idea is to take advantage of the independence among the constraints $\mathca
 \textcolor{DarkOrchid}{\textit{Signature Function.}} Let $f_i(\lambda) = \max\{(v_{E_i}-\lambda c_{E_i}) x | x\in \conv(\mathcal F_i) \}$ be the signature function of agent $i$. The signature function is piecewise-linar and convex.
 
 \textcolor{DarkOrchid}{\textit{Lagrangian Dual.}} The Lagrangian dual of LP\autoref{LP} is therefore
-\begin{equation}
+\begin{equation*}
 \label{eq:Lagrangiandual}
 \begin{aligned}
 \min_{\lambda} \left( B\lambda+\sum_i f_i(\lambda)\right).
 \end{aligned}
-\end{equation}
+\end{equation*}
 
 \begin{theorem}[4]\large
 $\tau(B)$ is piecewise-linear and concave.