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Yu Cong
2025-08-23 13:30:09 +08:00
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poster.tex
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@@ -56,14 +56,14 @@ We consider the incentive allocation problem with additional constraints.
} }
This problem is NP-hard. Consider its LP relaxation. This problem is NP-hard. Consider its LP relaxation.
\begin{equation}\label{LP} \begin{equation*}\label{LP}
\begin{aligned} \begin{aligned}
\tau(B)=\max_x&\; & v\cdot x& & & \\ \tau(B)=\max_x&\; & v\cdot x& & & \\
s.t.&\; & c \cdot x &\leq B & &\\ s.t.&\; & c \cdot x &\leq B & &\\
& & x_{E_i}&\in \conv(\mathcal{F}_i) & &\;\forall i\in [n]\\ & & x_{E_i}&\in \conv(\mathcal{F}_i) & &\;\forall i\in [n]\\
& & x&\in [0,1]^m & & & & x&\in [0,1]^m & &
\end{aligned} \end{aligned}
\end{equation} \end{equation*}
\textbf{Output}: The entire curve $\tau(B)$ for $B\in [0,\infty)$. \textbf{Output}: The entire curve $\tau(B)$ for $B\in [0,\infty)$.
We consider 3 cases of additional constraints $x_{E_i}\in \mathcal{F}_i$ : 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{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 \textcolor{DarkOrchid}{\textit{Lagrangian Dual.}} The Lagrangian dual of LP\autoref{LP} is therefore
\begin{equation} \begin{equation*}
\label{eq:Lagrangiandual} \label{eq:Lagrangiandual}
\begin{aligned} \begin{aligned}
\min_{\lambda} \left( B\lambda+\sum_i f_i(\lambda)\right). \min_{\lambda} \left( B\lambda+\sum_i f_i(\lambda)\right).
\end{aligned} \end{aligned}
\end{equation} \end{equation*}
\begin{theorem}[4]\large \begin{theorem}[4]\large
$\tau(B)$ is piecewise-linear and concave. $\tau(B)$ is piecewise-linear and concave.