Revert "remove begin equations"

This reverts commit 6fd167c.

Author: mmcky

lectures/match_transport.md

@@ -57,11 +57,13 @@ Given a *cost function* $c:X \times Y \rightarrow \mathbb{R}$, the (discrete) *o
 
 
 $$
+\begin{equation} 
 \begin{aligned}
 \min_{\mu \geq 0}& \sum_{(x,y) \in X \times Y} \mu_{xy}c_{xy}\\
 \text{s.t. }& \sum_{x \in X} \mu_{xy} = n_x \\
 & \sum_{y \in Y} \mu_{xy} = m_y 
 \end{aligned}
+\end{equation}
 $$
 
 Given our discreteness  assumptions about $n$ and $m$, the problem admits an integer solution $\mu \in \mathbb{Z}_+^{X \times Y}$, i.e. $\mu_{xy}$ is a non-negative integer for each $x\in X, y\in Y$.
@@ -99,11 +101,13 @@ In the following implementation we assume that the cost function is $c_{xy} = |x
 Hence, our problem is
 
 $$
+\begin{equation} 
 \begin{aligned}
 \min_{\mu \in \mathbb{Z}_+^{X \times Y}}& \sum_{(x,y) \in X \times Y} \mu_{xy}|x-y|^{1/\zeta}\\
 \text{s.t. }& \sum_{x \in X} \mu_{xy} = n_x \\
 & \sum_{y \in Y} \mu_{xy} = m_y 
 \end{aligned}
+\end{equation}
 $$
 
 The following class takes as inputs sets of types $X,Y \subset \mathbb{R},$ marginals $n, m $ with positive integer entries such that $\sum_{x \in X} n_x = \sum_{y \in Y} m_y $ and cost parameter $\zeta>1$.
@@ -1551,11 +1555,13 @@ example_3.plot_matching(matching_NAM, title = 'NAM', figsize = (5,5), add_labels
 Let us recall our formulation
 
 $$
+\begin{equation} 
  \begin{aligned}
 V_P = \min_{\mu \geq 0}& \sum_{(x,y) \in X \times Y} \mu_{xy}c_{xy}\\
 \text{s.t. }& \sum_{x \in X} \mu_{xy} = n_x \\
 & \sum_{y \in Y} \mu_{xy} = m_y 
 \end{aligned}
+\end{equation}
 $$
 
 The *dual problem* is 
@@ -1578,18 +1584,21 @@ Since the dual is feasible and bounded,  $V_P = V_D$ (*strong duality* prevails)
 Assume now that $y_{xy} = \alpha_x + \gamma_y - c_{xy}$ is the output generated by matching $x$ and $y.$ It includes the sum of $x$ and $y$ specific amenities/outputs minus the cost $c_{xy}.$ Then, we have can formulate the following problem and its dual
 
 $$
+\begin{equation*} 
  \begin{aligned}
 W_P = \max_{\mu \geq 0}& \sum_{(x,y) \in X \times Y} \mu_{xy}y_{xy}\\
 \text{s.t. }& \sum_{x \in X} \mu_{xy} = n_x \\
 & \sum_{y \in Y} \mu_{xy} = m_y 
 \end{aligned}
+\end{equation*} 
 $$
 
-$$
+$$ \begin{equation} 
 \begin{aligned}
- W_D = \min_{u,v}& \sum_{x \in X }n_x u_x + \sum_{y \in Y} m_y v_y\\
+ = W_D = \min_{u,v}& \sum_{x \in X }n_x u_x + \sum_{y \in Y} m_y v_y\\
 \text{s.t. }&  u_x + v_y \geq y_{xy} \\
 \end{aligned}
+\end{equation}
 $$