Fix latex for bilevel_introduction notebook

Author: marcos-dv

authors/eduardosalaz/bilevel/bilevel_introduction.ipynb

@@ -2,7 +2,9 @@
   "cells": [
     {
       "cell_type": "markdown",
-      "metadata": {},
+      "metadata": {
+        "id": "9_qd3IFVeXZB"
+      },
       "source": [
         "# Bilevel Optimization Introduction\n",
         "[![bilevel_introduction.ipynb](https://img.shields.io/badge/github-%23121011.svg?logo=github)](https://github.com/ampl/colab.ampl.com/blob/master/authors/eduardosalaz/bilevel/bilevel_introduction.ipynb) [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/ampl/colab.ampl.com/blob/master/authors/eduardosalaz/bilevel/bilevel_introduction.ipynb) [![Open In Deepnote](https://deepnote.com/buttons/launch-in-deepnote-small.svg)](https://deepnote.com/launch?url=https://github.com/ampl/colab.ampl.com/blob/master/authors/eduardosalaz/bilevel/bilevel_introduction.ipynb) [![Open In Kaggle](https://kaggle.com/static/images/open-in-kaggle.svg)](https://kaggle.com/kernels/welcome?src=https://github.com/ampl/colab.ampl.com/blob/master/authors/eduardosalaz/bilevel/bilevel_introduction.ipynb) [![Open In Gradient](https://assets.paperspace.io/img/gradient-badge.svg)](https://console.paperspace.com/github/ampl/colab.ampl.com/blob/master/authors/eduardosalaz/bilevel/bilevel_introduction.ipynb) [![Open In SageMaker Studio Lab](https://studiolab.sagemaker.aws/studiolab.svg)](https://studiolab.sagemaker.aws/import/github/ampl/colab.ampl.com/blob/master/authors/eduardosalaz/bilevel/bilevel_introduction.ipynb) [![Powered by AMPL](https://h.ampl.com/https://github.com/ampl/colab.ampl.com/blob/master/authors/eduardosalaz/bilevel/bilevel_introduction.ipynb)](https://ampl.com)\n",
@@ -22,7 +24,9 @@
     {
       "cell_type": "code",
       "execution_count": 1,
-      "metadata": {},
+      "metadata": {
+        "id": "r8Q6q4B_eXZE"
+      },
       "outputs": [],
       "source": [
         "# Install dependencies\n",
@@ -32,7 +36,9 @@
     {
       "cell_type": "code",
       "execution_count": 2,
-      "metadata": {},
+      "metadata": {
+        "id": "zEw9OOW7eXZE"
+      },
       "outputs": [],
       "source": [
         "# Google Colab & Kaggle integration\n",
@@ -69,7 +75,8 @@
         "& y^* \\in \\arg\\min_{y} \\{f(x,y) : g(x,y) \\leq 0\\} && \\text{(Follower's problem)}\n",
         "\\end{align}\n",
         "$\n",
-        "**​Key point**: The leader's objective depends on $y*$, which is the optimal solution to the follower's problem (which itself depends on $x$).\n",
+        "\n",
+        "**Key point**: The leader's objective depends on $y*$, which is the optimal solution to the follower's problem (which itself depends on $x$).\n",
         "\n",
         "## Why \"Bilevel\"?\n",
         "\n",
@@ -410,6 +417,7 @@
         "### Why Complementarity Represents \"Either/Or\" Logic\n",
         "\n",
         "The lower level problem is:\n",
+        "\n",
         "$$\n",
         "\\min_{y} \\quad (y - x)^2 \\quad \\text{s.t.} \\quad y \\geq 0\n",
         "$$\n",
@@ -459,11 +467,13 @@
         "#### **Case 2: Boundary Solution ($y = 0$, so $\\lambda > 0$)**\n",
         "\n",
         "If $y = 0$, from stationarity:\n",
+        "\n",
         "$$2(0 - x) - \\lambda = 0 \\implies \\lambda = -2x$$\n",
         "\n",
         "For dual feasibility, we need $\\lambda \\geq 0$, which requires $x \\leq 0$.\n",
         "\n",
         "Substituting $y = 0$ into the upper level:\n",
+        "\n",
         "$$\\min_{x \\leq 0} \\quad x^2 + (0 - 10)^2 = x^2 + 100$$\n",
         "\n",
         "This is minimized at $x = 0$ (closest feasible point to unconstrained minimum).\n",