Commit c2613d15 by Niklas Rieken

### typo

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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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