typos

Gurobi · Nov 14, 2024 · 6adcd71 · 6adcd71
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diff --git a/optimization202/Modeling_Session_1/completed_exercise_set1.ipynb b/optimization202/Modeling_Session_1/completed_exercise_set1.ipynb
@@ -98,7 +98,7 @@
     "\\end{align*}\n",
     "\n",
     "`Solution:`\n",
-    "Select $x = y = z = 1$\n",
+    "An example solution is to select $x = y = z = 1$.\n",
     "\n",
     "> **Q8b.** \n",
     "\n",
@@ -110,14 +110,22 @@
     "\n",
     "`Solution:`\n",
     "\n",
-    "Substitute $(1)$ into $(2)$:\n",
-    "$3x_1 + 2x_2 + x_3 = 2x_1 + x_2 + (x_1 + x_2 + x_3) = 2x_1 + x_2 + 6$\n",
+    "One way to derive a contradiction:\n",
+    "Multiply $(1)$ by 3, which gives $3x_1 + 3x_2 + 3x_3 = 18$. \n",
     "\n",
-    "This simplifies $(2)$ as $2x_1 + x_2 \\ge 14$.\n",
+    "Since $x_1, x_2, x_3$ are all nonnegative,\n",
+    "\\begin{align*}\n",
+    "3x_1 + 3x_2 + 3x_3 \\ge 3x_1 + 2x_2 + x_3 \\ge 20\n",
+    "\\end{align*}\n",
+    "\n",
+    "This reduces to:\n",
     "\n",
-    "Since $x_3$ is nonnegative, the first equation implies $x_1 + x_2\\le 6$, or $-x_1 - x_2 \\ge -6$.\n",
+    "\\begin{align*}\n",
+    "18 \\ge 3x_1 + &2x_2 + x_3 \\ge 20 \\\\\n",
+    "18 &\\ge 20\n",
+    "\\end{align*}\n",
     "\n",
-    "Adding the inequalities gives $x_1 \\ge 8$, which can't happen due to $(1)$. \n",
+    "Which is a contradiction. \n",
     "\n",
     "**Q9.** Translate the following into constraints using binary variables for deciding to invest in three stocks. Then determine if they are compatible as above. \n",
     "\n",

diff --git a/optimization202/Modeling_Session_1/completed_nonlinear_regression_exercise.ipynb b/optimization202/Modeling_Session_1/completed_nonlinear_regression_exercise.ipynb
@@ -63,7 +63,7 @@
     "\n",
     "\n",
     "\n",
-    "In many applicaiton such problems are solved using a reduced gradient method that cannot garantee global optimality. In nonlienar regression, it is possible to have multiple local optima because they are typically nonconvex. "
+    "In many application such problems are solved using a reduced gradient method that cannot guarantee global optimality. In nonlinear regression, it is possible to have multiple local optima because they are typically nonconvex. "
    ]
   },
   {

diff --git a/optimization202/Modeling_Session_1/nonlinear_regression_exercise.ipynb b/optimization202/Modeling_Session_1/nonlinear_regression_exercise.ipynb
@@ -63,7 +63,7 @@
     "\n",
     "\n",
     "\n",
-    "In many applicaiton such problems are solved using a reduced gradient method that cannot garantee global optimality. In nonlienar regression, it is possible to have multiple local optima because they are typically nonconvex. "
+    "In many application such problems are solved using a reduced gradient method that cannot guarantee global optimality. In nonlinear regression, it is possible to have multiple local optima because they are typically nonconvex. "
    ]
   },
   {