diff --git a/poster.tex b/poster.tex index e3c833c..3f45614 100644 --- a/poster.tex +++ b/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.