From 886a0623a321c4179af253027f83a9365df4d1f4 Mon Sep 17 00:00:00 2001
From: Orestis Malaspinas <orestis.malaspinas@hesge.ch>
Date: Sun, 25 Nov 2018 23:00:27 +0100
Subject: [PATCH] adde local minimum and ackley function

---
 cours.md         |  23 +-
 figs/ackley.svg  | 757 +++++++++++++++++++++++++++++++++++++++++++++++
 python/ackley.py |  10 +
 3 files changed, 781 insertions(+), 9 deletions(-)
 create mode 100644 figs/ackley.svg
 create mode 100644 python/ackley.py

diff --git a/cours.md b/cours.md
index b57d88e..65a5280 100644
--- a/cours.md
+++ b/cours.md
@@ -1578,22 +1578,27 @@ Il existe comme on vient de le voir deux types disctincts de problèmes d'optimi
 
 ### L'optimisation continue
 
-Soit $f:\real^n\rightarrow\real$ une fonction objectif, on cherche $\vec x_0\in\real^n$, tel que $f(\vec x_0)\leq f(\vec x)$ pour $\vec x$ certaines conditions.
-Les conditions peuvent se séparer en deux parties différentes, les contraintes d'égalités et d'inégalités.
-Soient $m$ fonctions $g_i:\real^n\rightarrow\real$ et $n$ fonctions $h_j:\real^n\rightarrow$
+Soit $f:\real^n\rightarrow\real$ une fonction objectif (ou fontion de coût), on cherche $\vec x_0\in\real^n$, tel que $f(\vec x_0)\leq f(\vec x)$ pour $\vec x$ certaines conditions: les contraintes qui sont en général des égalités strictes ou des inégalités qui peuvent s'exprimer de la façon suivante.
+Soient $m$ fonctions $g_i:\real^n\rightarrow\real$
 \begin{align}
-&g_i(\vec x)\leq 0,\quad i=1,...,m,\mbox{ et}\\
-&h_j(\vec x)= 0,\quad j=1,...,n.
+&g_i(\vec x)\leq 0,\quad i=1,...,m.
 \end{align}
-Si $m=0$ et $n=0$ on a à faire à un problème d'optimisation sans contraintes. On peut résumer tout cela comme
+Si $m=0$ on a à faire à un problème d'optimisation sans contraintes. On peut résumer tout cela comme
 \begin{align*}
 &\min_{\vec x\in\real^n}f(\vec x),\\
-&g_i(\vec x)\leq 0,\quad i=1,...,m,\mbox{ et}\\
-&h_j(\vec x)= 0,\quad j=1,...,n,\\
-&m\geq 0,\quad n\geq 0.
+&g_i(\vec x)\leq 0,\quad i=1,...,m,\\
+&\mbox{pour }m\geq 0.
 \end{align*}
+Les contraintes limites l'espace des solutions et forment un sous-ensemble, noté $A$, de $\real^n$ ($A\subseteq\real^n$).
 
+Une des difficultés pour déterminer le minimum d'une fonction coût est l'existence de plusieurs minima locaux.
+Un **minimum local**, $\vec x^\ast\in A$, est tel que pour une région proche de $\vec x^\ast$, on a que $f(\vec x)\geq f(\vec x^\ast)$. Une fonction particulièrement problématique où il est
+très difficile de trouver le minimum global est une fonction de Ackley. En une dimension, elle est de la forme (voir la @fig:ackley)
+$$
+f(x)=-20e^{-0.2*\sqrt{0.5x^2}}-e^{0.5(\cos(2\pi x))}+e+20.
+$$
 
+![La fonction d'Ackley en une dimension. Il est particulièrement compliqué de trouver le minimum global en raison de l'existence d'une multitude de minima locaux.](figs/ackley.svg){#fig:ackley width=70%}
 
 [^1]: On pourrait, de façon similaire, utiliser la formule de différences finies en avant ou en arrière (ou un mélange des deux).
 [^2]: Comme ce polynôme passe par les points $(x_0,y_0)$, $(x_1,y_1)$, ..., $(x_m,y_m)$, il est unique, c'est donc exactement le même que celui exprimé avec les $\{a_i\}_{i=0}^m$.
diff --git a/figs/ackley.svg b/figs/ackley.svg
new file mode 100644
index 0000000..5f1a9a9
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@@ -0,0 +1,757 @@
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diff --git a/python/ackley.py b/python/ackley.py
new file mode 100644
index 0000000..6445ae4
--- /dev/null
+++ b/python/ackley.py
@@ -0,0 +1,10 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+N=30
+x = np.arange(-30,30+1/N,1/N)
+
+y = -20*np.exp(-0.2*np.sqrt(0.5*x**2))-np.exp(0.5*(np.cos(2*np.pi*x)))+np.exp(1)+20
+
+plt.plot(x,y)
+plt.show()
\ No newline at end of file
-- 
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