Retained n = 4 example demo

examples/04/example.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Julian's Small Example\n",
+    "\n",
+    "Julian came up with a small $n = 2$ example (excluding sex data) for testing the theory behind solving the problem. This example can easily be extended and then solved using the AlphaRGS module. This is a quick demonstration of using the module within an interactive notebook environment."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import alphargs"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Problem Variables\n",
+    "\n",
+    "The variables for Julian's $n = 2$ problem are as follows:\n",
+    "$$\n",
+    "    \\Sigma = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix},\\quad\\\n",
+    "    \\bar{\\mu} = \\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix},\\quad\\\n",
+    "    \\Omega = \\begin{bmatrix} \\frac{1}{9} & 0 \\\\ 0 & 4 \\end{bmatrix}.\n",
+    "$$\n",
+    "This can easily be extended to include sex data by reflecting those variable across both sires and dams, filling in the remainder with zeros. \n",
+    "$$\n",
+    "    \\Sigma = \\begin{bmatrix}\n",
+    "                1 & 0 & 0 & 0 \\\\\n",
+    "                0 & 1 & 0 & 0 \\\\\n",
+    "                0 & 0 & 1 & 0 \\\\\n",
+    "                0 & 0 & 0 & 1\n",
+    "             \\end{bmatrix},\\quad\\\n",
+    "    \\bar{\\mu} = \\begin{bmatrix} 1 \\\\ 1 \\\\ 2 \\\\ 2  \\end{bmatrix},\\quad\\\n",
+    "    \\Omega = \\begin{bmatrix}\n",
+    "                \\frac{1}{9} & 0 & 0 & 0 \\\\\n",
+    "                0 & \\frac{1}{9} & 0 & 0 \\\\\n",
+    "                0 & 0 & 4 & 0 \\\\\n",
+    "                0 & 0 & 0 & 4\n",
+    "             \\end{bmatrix},\\quad\\\n",
+    "    \\mathcal{S} = \\lbrace 1, 3 \\rbrace,\\quad\\\n",
+    "    \\mathcal{D} = \\lbrace 2, 4 \\rbrace.\n",
+    "$$\n",
+    "The matrix variables are stored in [`A04.txt`](A04.txt), [`EBV04.txt`](EBV04.txt), and [`S04.txt`](S04.txt) respectively, where the first and last are matrices in coordinate format. These are then loaded into Python using `load_problem`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# key problem variables loaded from standard format txt files\n",
+    "sigma, mubar, omega, n = alphargs.load_problem(\"A04.txt\", \"EBV04.txt\", \"S04.txt\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Since we have constructed the problem to have alternating sex data, we may approach it as we did in [`example.py`](../50/example.py) using range iterates."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "sires = range(0, n, 2)\n",
+    "dams = range(1, n, 2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Gurobi\n",
+    "\n",
+    "AlphaRGS defines functions for solving the standard and robust genetic selection problems using the [gurobipy](https://pypi.org/project/gurobipy/) Python interface to Gurobi. Below these are used to solve the above problem for $\\lambda = 0.5, \\kappa = 1$: "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Set parameter Username\n",
+      "Academic license - for non-commercial use only - expires 2025-02-26\n",
+      "i  w_std    w_rbs\n",
+      "1  0.00000  0.38200\n",
+      "2  0.00000  0.38200\n",
+      "3  0.50000  0.11800\n",
+      "4  0.50000  0.11800\n",
+      "\n",
+      "w_std objective: 1.87500\n",
+      "w_rbs objective: 0.77684 (z = 0.37924)\n",
+      "Maximum change: 0.38200\n",
+      "Average change: 0.38200\n",
+      "Minimum change: 0.38200\n"
+     ]
+    }
+   ],
+   "source": [
+    "lam = 0.5\n",
+    "kap = 1\n",
+    "\n",
+    "# computes the standard and robust genetic selection solutions\n",
+    "w_std, obj_std = alphargs.gurobi_standard_genetics(sigma, mubar, sires, dams, lam, n)\n",
+    "w_rbs, z_rbs, obj_rbs = alphargs.gurobi_robust_genetics(sigma, mubar, omega, sires, dams, lam, kap, n)\n",
+    "\n",
+    "# print a comparison of the two solutions\n",
+    "alphargs.print_compare_solutions(w_std, w_rbs, obj_std, obj_rbs, z2=z_rbs, name1=\"w_std\", name2=\"w_rbs\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Finally, we can use `check_uncertainty_constraint` to explore how close our $z\\geq \\sqrt{w^{T}\\Omega w}$ constraint came to equality."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "     z: 0.37923871642022844\n",
+      "w'*Ω*w: 0.3792386953366983\n",
+      "  Diff: 2.1083530143961582e-08\n"
+     ]
+    }
+   ],
+   "source": [
+    "if not alphargs.check_uncertainty_constraint(z_rbs, w_rbs, omega, debug=True):\n",
+    "    raise ValueError"
+   ]
+  }
+ ],
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