Updated and published constraint qualifications notes

ndcbe · Nov 26, 2024 · 5c7fd0b · 5c7fd0b
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diff --git a/notebooks/7-dev/Constraint-Qualifications.ipynb b/notebooks/7-dev/Constraint-Qualifications.ipynb
@@ -65,7 +65,7 @@
     "\n",
     "\\begin{equation*}\n",
     "\n",
-    "\\text{KKT conditions (linearization,~first~order)} \\xleftrightarrow{\\text{constraint~qualifications}} \\text{Theorem~4.8}\n",
+    "\\text{KKT conditions (linearization, first order)} \\leftarrow \\text{constraint qualifications} \\rightarrow \\text{Theorem~4.8}\n",
     "\\end{equation*}\n",
     "\n",
     "Constraint qualification act as the link between KKT conditions (which we numerical solve) and optimization theory.\n",
@@ -214,16 +214,19 @@
     "\n",
     "\n",
     "The Lagrangian is defined as:\n",
+    "\n",
     "$$\n",
     "L(x,u) = f(x) + g(x)^T u\n",
     "$$\n",
     "\n",
     "Specifically:\n",
+    "\n",
     "$$\n",
     "L = x_1 + u_1 (x_2 - x_1^3) + u_2 (-x_1^3 - x_2)\n",
     "$$\n",
     "\n",
     "Set the gradient of the Lagrangian with respect to $x$ equal to zero:\n",
+    "\n",
     "$$\n",
     "\\nabla_x L = \n",
     "\\begin{bmatrix}\n",
@@ -261,6 +264,7 @@
     "   $u_1 = 0$, $u_2 = 0$\n",
     "\n",
     "   Gradient of the Lagrangian:\n",
+    "   \n",
     "   $$\n",
     "   \\nabla_x L = \n",
     "   \\begin{bmatrix}\n",
@@ -273,12 +277,14 @@
     "   0\n",
     "   \\end{bmatrix}\n",
     "   $$\n",
-    "   **Solution is not bounded!**\n",
+    "   \n",
+    "   *Solution is not bounded!*\n",
     "\n",
     "2. **$g_1$ is active, $g_2$ is NOT strongly active**  \n",
     "   $u_1 \\geq 0$, $u_2 = 0$\n",
     "\n",
     "   Gradient of the Lagrangian:\n",
+    "   \n",
     "   $$\n",
     "   \\nabla_x L = \n",
     "   \\begin{bmatrix}\n",
@@ -292,7 +298,7 @@
     "   \\end{bmatrix}\n",
     "   $$\n",
     "\n",
-    "   From this, $u_1 = 0$... **KKT conditions are not satisfied!**\n",
+    "   From this, $u_1 = 0$... *KKT conditions are not satisfied!*\n",
     "\n",
     "3. **$g_1$ is NOT strongly active, $g_2$ is active**  \n",
     "   Same problem as case 2.\n",
@@ -301,6 +307,7 @@
     "   $u_1 \\geq 0$, $u_2 \\geq 0$\n",
     "\n",
     "   Gradient of the Lagrangian:\n",
+    "\n",
     "   $$\n",
     "   \\nabla_x L = \n",
     "   \\begin{bmatrix}\n",
@@ -317,16 +324,18 @@
     "   From $\\nabla_{x_2} L = 0$, we get $u_1 = u_2$.\n",
     "\n",
     "   Substituting into $\\nabla_{x_1} L = 0$:\n",
+    "\n",
     "   $$\n",
     "   1 - 3u_1 x_1^2 - 3u_2 x_1^2 = 0\n",
     "   $$\n",
     "\n",
     "   Solving for $x_1$:\n",
+    "   \n",
     "   $$\n",
     "   x_1 = \\sqrt{\\frac{1}{3u_1}}\n",
     "   $$\n",
     "\n",
-    "   This means we can see from visual inspection that the **optimal solution occurs at**:\n",
+    "   This means we can see from visual inspection that the *optimal solution occurs at*:\n",
     "   - $x_1 \\to 0$, $x_2 \\to 0$\n",
     "   - $u_1, u_2 \\to \\infty$\n",
     "\n",
@@ -381,8 +390,22 @@
     "Recall both constraints are active at the solution $x_1 = x_2 = 0$:\n",
     "\\begin{equation*}\n",
     "\\begin{bmatrix}\n",
-    "\\sout{-3x_1^2}~{0} & 1 \\\\\n",
-    "\\sout{-3x_1^2}~{0} & -1\n",
+    "-3x_1^2 & 1 \\\\\n",
+    "-3x_1^2 & -1\n",
+    "\\end{bmatrix}\n",
+    "\\cdot d = \n",
+    "\\begin{bmatrix}\n",
+    "0 \\\\\n",
+    "0\n",
+    "\\end{bmatrix}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "Substituting $x_1 = x_2 = 0$:\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\begin{bmatrix}\n",
+    "0 & 1 \\\\\n",
+    "0 & -1\n",
     "\\end{bmatrix}\n",
     "\\cdot d = \n",
     "\\begin{bmatrix}\n",