Commit c2613d15 authored by Niklas Rieken's avatar Niklas Rieken

typo

parent 99d588b3
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......@@ -12,8 +12,12 @@
\usepackage{float}
\usepackage{commath}
\usepackage{tikz}
\usepackage{pgfplots}
\usepgfplotslibrary{patchplots}
\pgfplotsset{compat=newest}
\pgfplotsset{plot coordinates/math parser=false}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc}
\usetikzlibrary{calc,intersections}
\usepackage{listings}
\usepackage{enumitem}
\usepackage{algorithm2e}
......@@ -77,13 +81,38 @@ We distinguish (for now) three classes od optimization problems:
\end{description}
Note that every LP is a convex optimization problem and every convex optimization problem is an NLP. In particular, every LP is also an NLP.
Intuitively $\alpha a + \beta b$ with $\alpha + \beta = 1$ and $\alpha, \beta \geq 0$ is a straight line segment between $a$ and $b$: $\alpha a + \beta b = (1-\beta)a + \beta b = a + \beta(b-a)$.
\begin{figure}
%TODO straight line
Intuitively, $\alpha a + \beta b$ with $\alpha + \beta = 1$ and $\alpha, \beta \geq 0$ is a straight line segment between $a$ and $b$: $\alpha a + \beta b = (1-\beta)a + \beta b = a + \beta(b-a)$.
\begin{figure}[H]
\centering
\begin{tikzpicture}
\path (0, 0) coordinate(a) -- (2, 1) coordinate(b);
\draw[-] (a) -- (b) node[pos=-.1] {$a$} node[pos=1.1] {$b$};
\draw[->,shorten >= .2cm] (.2, 0) -- +($(b) - (a)$) node[pos=.5, below right] {\scriptsize$(b-a)$};
\end{tikzpicture}
\end{figure}
Hence, the condition for a function $f$ to be convex is that any straight line segment between two arbitrary points $(a, f(a)), (b, f(b)$ is above every point of the function between $a$ and $b$.
\begin{figure}
%TODO convex and non-convex function
Hence, the condition for a function $f$ to be convex is that any straight line segment between two arbitrary points $(a, f(a)), (b, f(b))$ is above every point of the function between $a$ and $b$.
\begin{figure}[H]
\centering
%\begin{tikzpicture}
% \begin{axis}[axis y line=left, xmin=0, xmax=5, xlabel=$x$, ylabel=$f(x)$, ymin=-3.5, ymax=2, xtick={\empty}, ytick={\empty}, axis on top, axis x line=bottom, every axis plot post/.append style={mark=none,domain=0:5,samples=50,smooth}]
% \addplot{x^2 - 5*x + 4};
% \end{axis}
%\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[width=5in,axis equal image,
axis lines=middle,
xmin=0,xmax=8,
xlabel=$x$,ylabel=$f(x)$,
ymin=0,ymax=4,
xtick={\empty},ytick={\empty}, axis on top
]
\addplot[thick,domain=0.25:7,blue,name path=A] {-x/3 + 2.75} coordinate[pos=0.4] (m) ;
\draw[thick,blue,name path=B] (0.15,4) .. controls (1,1) and (4,0) .. (6,2) coordinate[pos=0.075] (a1) coordinate[pos=0.95] (a2);
\path[name intersections={of=A and B, by={a,b}}];
\draw[densely dashed] (0,0) -| node[pos=0.5, color=black, below] {$a$}(a);
\draw[densely dashed] (0,0) -| node[pos=0.5, color=black, below] {$b$}(b);
\end{axis}
\end{tikzpicture}
\end{figure}
\section{Theory}
......
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