misc

Author: jstac

lectures/lp_intro.md

@@ -28,11 +28,11 @@ If a primal problem involves **maximization**, the dual problem involves **minim
 
 If a primal problem involves  **minimization**, the dual problem involves **maximization**.
 
-We provide a standard form of a  linear program and methods to transform other forms of linear programming problems  into a standard form.
+We provide a standard form of a linear program and methods to transform other forms of linear programming problems  into a standard form.
 
-We tell how to solve a linear programming prolem using **Scipy**.
+We tell how to solve a linear programming problem using [SciPy](https://scipy.org/).
 
-We describe the important concept of  complementary slackness and how it relates to the dual problem.
+We describe the important concept of complementary slackness and how it relates to the dual problem.
 
 Let's start with some standard imports.
 
@@ -436,9 +436,9 @@ Inside it, Python first  transforms the problem into  standard form.
 
 To do that, for each inequality constraint it generates one slack variable.
 
-Here the vector of slack variables is a two-dimensional numpy array that  equals $b_{ub} - A_{ub}x$. 
+Here the vector of slack variables is a two-dimensional NumPy array that  equals $b_{ub} - A_{ub}x$. 
 
-See [official documentation](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.linprog.html#scipy.optimize.linprog) for more details.
+See the [official documentation](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.linprog.html#scipy.optimize.linprog) for more details.
 
 ```{note}
 This problem is to maximize the objective, so that we need to put a minus sign in front of parameter vector c.
@@ -801,4 +801,4 @@ As for numerical results,
 
 3. $p_6 = -0.16$, which means one dollar of the loan amount at the beginning of the third year is worth $\$ 0.16$. Since $|p_6|$ is higher than the interest rate 6\%, the mutual fund should borrow as much as possible at the beginning of the third year. Recall that  the optimal solution to the primal problem is  $x_4 = -20,000$ which means the mutual fund borrows money from the bank as much as it can.
 
-4. $p_7 = 0.0015$, which means one dollar of the amount of the corporate bond that the mutual fund can buy is worth $\$ 0.0015$.
+4. $p_7 = 0.0015$, which means one dollar of the amount of the corporate bond that the mutual fund can buy is worth $\$ 0.0015$.