From c49884054c55db5120ece5c06e1db63c978fa7be Mon Sep 17 00:00:00 2001
From: "J. Gergaud" <gergaud@ap240726-01.local>
Date: Sun, 24 Nov 2024 23:58:19 +0100
Subject: [PATCH] foo

---
 M2/TP-introduction-NN.ipynb              | 128 ++++++++++++++++++
 M2/TP-steepest-descent-quad.ipynb        | 158 +++++++++++++++++++++++
 "M2/nn \342\200\223 Cours.pdf"           | Bin 0 -> 46917 bytes
 "M2/optimization \342\200\223 Cours.pdf" | Bin 236533 -> 273510 bytes
 4 files changed, 286 insertions(+)
 create mode 100644 M2/TP-introduction-NN.ipynb
 create mode 100644 M2/TP-steepest-descent-quad.ipynb
 create mode 100644 "M2/nn \342\200\223 Cours.pdf"

diff --git a/M2/TP-introduction-NN.ipynb b/M2/TP-introduction-NN.ipynb
new file mode 100644
index 0000000..6ee9dd0
--- /dev/null
+++ b/M2/TP-introduction-NN.ipynb
@@ -0,0 +1,128 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Neural Network\n",
+    "\n",
+    "Complete the descent with constant rate function"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "julia"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "function my_descent(f,x0::Vector{<:Real};η=0.1,AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)# to complete\n",
+    "\n",
+    "  return xₖ, flag, fₖ, ∇fₖ, k  c\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "And apply this code for solving the following problem\n",
+    "\n",
+    "$$(P)\\left\\{\\begin{array}{l}\n",
+    "Min\\;f(\\beta) = \\frac{1}{n}\\|X\\beta - y\\|^2\\\\\n",
+    "\\beta\\in\\R^2\n",
+    "\\end{array}\\right.\n",
+    "$$\n",
+    "with\n",
+    "$$X = \\begin{pmatrix}\n",
+    "1 & x_1\\\\\n",
+    "\\vdots & \\vdots\\\\\n",
+    "1 & x_5\n",
+    "\\end{pmatrix}\n",
+    "\\;\\;\\textrm{and}\\;\\;\n",
+    "y = \\begin{pmatrix}\n",
+    "y_1\\\\\n",
+    "\\vdots\\\\\n",
+    "y_5\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "You 'll print the first 7 iterations for the initual guess \n",
+    "$x0= (n/2)(9,1)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "plaintext"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "using Plots\n",
+    "# n = 5\n",
+    "x1 = 1.; x2 = sqrt(5*9/2-x1^2);\n",
+    "x = [-x2,-x1,0.,x1,x2]\n",
+    "n = length(x)\n",
+    "a₁ = 1; a₀ = 2\n",
+    "y = a₁*x .+ a₀ # model\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "julia"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "function descent_backtrac(f,x0::Vector{<:Real};AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)\n",
+    "    # to complete\n",
+    "\n",
+    "  return xₖ, flag, fₖ, ∇fₖ, k  c\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "julia"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "using Plots\n",
+    "x = range(-10*(n/2)-xsol[1],stop=10*(n/2)-xsol[1],length=100)\n",
+    "y = range(-9*(n/2)-xsol[2],stop=9*(n/2)-xsol[2],length=100)\n",
+    "f_contour(x,y) = f([x,y])\n",
+    "z = @. f_contour(x', y)\n",
+    "nb_levels = 7\n",
+    "x0 = (n/2)*[9,1]"
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/M2/TP-steepest-descent-quad.ipynb b/M2/TP-steepest-descent-quad.ipynb
new file mode 100644
index 0000000..fb8a13c
--- /dev/null
+++ b/M2/TP-steepest-descent-quad.ipynb
@@ -0,0 +1,158 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Steepest descent for a quadratic function\n",
+    "$f(x) = x^Ax + b^Tx + c$, where $A$ is symetric and positive-definite\n",
+    "\n",
+    "Complete the steepest_descent_quad function"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "julia"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "function steepest_descent_quad(A::Matrix{<:Real},b::Vector{<:Real},c::Real,x0::Vector{<:Real};AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)\n",
+    "# to complete\n",
+    "\n",
+    "  return xₖ, flag, fₖ, ∇fₖ, k  c\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "plaintext"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "using Plots\n",
+    "# n = 5\n",
+    "x1 = 1.; x2 = sqrt(5*9/2-x1^2);\n",
+    "x = [-x2,-x1,0.,x1,x2]\n",
+    "n = length(x)\n",
+    "X = [ones(n)  x]\n",
+    "a₁ = 1; a₀ = 2\n",
+    "y = a₁*x .+ a₀ # model\n",
+    "A = (1/n)*X'*X # A=[1 0 ; 0 9]\n",
+    "println(\"A = \", A)\n",
+    "b = -(2/n)*X'*y\n",
+    "c = (1/n)*y'*y\n",
+    "println(\"b = \", b)\n",
+    "println(\"n= \",n)\n",
+    "\n",
+    "f(x) = x'*A*x + b'*x + c \n",
+    "xsol = -(A' + A)\\b\n",
+    "xx = range(-10*(n/2)-xsol[1],stop=10*(n/2)-xsol[1],length=100)\n",
+    "yy = range(-9*(n/2)-xsol[2],stop=9*(n/2)-xsol[2],length=100)\n",
+    "f_contour(x,y) = f([x,y])\n",
+    "z = @. f_contour(xx', yy)\n",
+    "nb_levels = 7\n",
+    "x0 = (n/2)*[9,1]\n",
+    "xₖ = x0\n",
+    "Xsol = zeros(nb_levels+1,2)\n",
+    "Xsol[1,:] = x0\n",
+    "p1 = plot()\n",
+    "for nbit in 1:nb_levels\n",
+    "  xsol, flag, fsol, ∇f_xsol , nb_iter  = steepest_descent_quad(A,b,c,x0,nbit_max=nbit)\n",
+    "  plot!(p1,[xₖ[1],xsol[1]],[xₖ[2],xsol[2]],arrow=true)\n",
+    "  xₖ = xsol\n",
+    "  Xsol[nbit+1,:] = xsol\n",
+    "end\n",
+    "\n",
+    "levels = [f(Xsol[k,:]) for k in nb_levels+1:-1:1]\n",
+    "contour!(p1,xx,yy,z,levels=levels,cbar=false,color=:turbo)\n",
+    "\n",
+    "\n",
+    "xsol, flag, fsol, ∇f_xsol , nb_iter = algo_Newton(f,x0)\n",
+    "println()\n",
+    "println(\"Result with Newton's algorithm : \")\n",
+    "println(\"xsol = \", xsol)\n",
+    "println(\"flag = \", flag)\n",
+    "println(\"fsol = \", fsol)\n",
+    "println(\"∇f_xsol = \", ∇f_xsol)\n",
+    "println(\"nb_iter = \", nb_iter)\n",
+    "scatter!(p1,[xsol[1]],[xsol[2]])\n",
+    "plot!(p1,[x0[1],xsol[1]],[x0[2],xsol[2]],arrow=true)\n",
+    "plot(p1,legend=false)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "julia"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "function descent_backtrac(f,x0::Vector{<:Real};AbsTol= abs(eps()), RelTol = abs(eps()), ε=0.01, nbit_max = 0)\n",
+    "    # to complete\n",
+    "\n",
+    "  return xₖ, flag, fₖ, ∇fₖ, k  c\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "vscode": {
+     "languageId": "julia"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "using Plots\n",
+    "x = range(-10*(n/2)-xsol[1],stop=10*(n/2)-xsol[1],length=100)\n",
+    "y = range(-9*(n/2)-xsol[2],stop=9*(n/2)-xsol[2],length=100)\n",
+    "f_contour(x,y) = f([x,y])\n",
+    "z = @. f_contour(x', y)\n",
+    "nb_levels = 7\n",
+    "x0 = (n/2)*[9,1]\n",
+    "xₖ = x0\n",
+    "p1 = plot()\n",
+    "for nbit in 1:nb_levels\n",
+    "  xsol, flag, fsol, ∇f_xsol , nb_iter  = descent_backtrac(f,x0,nbit_max=nbit)\n",
+    "  #xsol, flag, fsol, ∇f_xsol , nb_iter  = my_descent(f,x0,nbit_max=nbit)\n",
+    "  println(\"xsol = \", xsol)\n",
+    "  println(\"flag = \", flag)\n",
+    "  println(\"fsol = \", fsol)\n",
+    "  println(\"∇f_xsol = \", ∇f_xsol)\n",
+    "  println(\"nb_iter = \", nb_iter)\n",
+    "  plot!(p1,[xₖ[1],xsol[1]],[xₖ[2],xsol[2]],arrow=true)\n",
+    "  xₖ = xsol\n",
+    "end\n",
+    "levels = [f(Xsol[k,:]) for k in nb_levels+1:-1:1]\n",
+    "contour!(p1,xx,yy,z,levels=levels,cbar=false,color=:turbo)\n",
+    "plot(p1,legend=false)"
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git "a/M2/nn \342\200\223 Cours.pdf" "b/M2/nn \342\200\223 Cours.pdf"
new file mode 100644
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