[match_transport] Fix notation typos in OT constraints and dual update (#304)

Co-authored-by: Humphrey Yang <39026988+HumphreyYang@users.noreply.github.com>

Author: sudayuga

lectures/match_transport.md

@@ -110,8 +110,8 @@ Hence, our problem is
 $$
 \begin{aligned}
 \min_{\mu \in \mathbb{Z}_+^{X \times Y}}& \sum_{(x,y) \in X \times Y} \mu_{xy}|x-y|^{1/\zeta} \\
-\text{s.t. }& \sum_{x \in X} \mu_{xy} = n_x \\
-& \sum_{y \in Y} \mu_{xy} = m_y 
+\text{s.t. }& \sum_{y \in Y} \mu_{xy} = n_x \\
+& \sum_{x \in X} \mu_{xy} = m_y 
 \end{aligned}
 $$
 
@@ -1677,8 +1677,8 @@ Let's recall the formulation
 $$
 \begin{aligned}
 V_P = \min_{\mu \geq 0}& \sum_{(x,y) \in X \times Y} \mu_{xy}c_{xy} \\
-\text{s.t. }& \sum_{x \in X} \mu_{xy} = n_x \\
-& \sum_{y \in Y} \mu_{xy} = m_y 
+\text{s.t. }& \sum_{y \in Y} \mu_{xy} = n_x \\
+& \sum_{x \in X} \mu_{xy} = m_y 
 \end{aligned}
 $$
 
@@ -1706,8 +1706,8 @@ Then we  can formulate the following problem and its dual
 $$
  \begin{aligned}
 W_P = \max_{\mu \geq 0}& \sum_{(x,y) \in X \times Y} \mu_{xy}y_{xy} \\
-\text{s.t. }& \sum_{x \in X} \mu_{xy} = n_x \\
-& \sum_{y \in Y} \mu_{xy} = m_y 
+\text{s.t. }& \sum_{y \in Y} \mu_{xy} = n_x \\
+& \sum_{x \in X} \mu_{xy} = m_y 
 \end{aligned}
 $$
 
@@ -1956,7 +1956,7 @@ exam_assign_OD.plot_hierarchies(subpairs)
 
 We proceed to describe and implement the algorithm to compute the dual solution. 
 
-As already mentioned, the algorithm starts from the matched pairs $(x_0,y_0)$ with no subpairs and assigns the (temporary) values $\psi_{x_0} = c_{x_0 y_0}$ and $\psi_{y_0} = 0,$ i.e. the $x$ type sustains the whole cost of matching. 
+As already mentioned, the algorithm starts from the matched pairs $(x_0,y_0)$ with no subpairs and assigns the (temporary) values $\phi_{x_0} = c_{x_0 y_0}$ and $\psi_{y_0} = 0,$ i.e. the $x$ type sustains the whole cost of matching. 
 
 
 
@@ -1976,7 +1976,7 @@ $$
 \leq \min (c_{x_0 y_j} + c_{x_i y_0} - c_{x_0 y_0} , c_{x_i y_j}) -  c_{x_j y_j} , \quad \text{for all } 1 \leq i < j \leq p.
 $$
 
-Then for all $i \in [p]$ compute the adjustment $ \Delta_i = \sum_{k = i+1}^p \beta_k + \phi_{x_p} - \phi_{x_1}$ and modify the dual variables 
+Then for all $i \in [p]$ compute the adjustment $ \Delta_i = \sum_{k = i+1}^p \beta_k + \phi_{x_p} - \phi_{x_i}$ and modify the dual variables 
 
 $$
 \begin{aligned}