commit 17c46636f9508aca353088cac0b6ebe4eaf49bb9 parent cdae52e238a65b7490caa996583df9fd4f3a9d58 Author: miksa <milutin@popovic.xyz> Date: Thu, 19 Jan 2023 16:21:08 +0000 nlin optimization done with sheet 1 Diffstat:
| A | nlin_opt/build/pics/sesh1_ex5.png | | | 0 | |
| A | nlin_opt/build/sesh1.pdf | | | 0 | |
| A | nlin_opt/build/sesh6.aux | | | 30 | ++++++++++++++++++++++++++++++ |
| A | nlin_opt/build/sesh6.fdb_latexmk | | | 248 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | nlin_opt/build/sesh6.fls | | | 1306 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | nlin_opt/build/sesh6.log | | | 1093 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | nlin_opt/build/sesh6.out | | | 7 | +++++++ |
| M | nlin_opt/build/sesh6.pdf | | | 0 | |
| A | nlin_opt/build/sesh6.toc | | | 7 | +++++++ |
| M | nlin_opt/preamble.tex | | | 2 | +- |
| A | nlin_opt/sesh1.tex | | | 329 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
11 files changed, 3021 insertions(+), 1 deletion(-)
diff --git a/nlin_opt/build/pics/sesh1_ex5.png b/nlin_opt/build/pics/sesh1_ex5.png Binary files differ. diff --git a/nlin_opt/build/sesh1.pdf b/nlin_opt/build/sesh1.pdf Binary files differ. diff --git a/nlin_opt/build/sesh6.aux b/nlin_opt/build/sesh6.aux @@ -0,0 +1,30 @@ +\relax +\providecommand\zref@newlabel[2]{} +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\@writefile{toc}{\contentsline {section}{\numberline {1}Sheet 6}{1}{section.1}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Exercise 35}{1}{subsection.1.1}\protected@file@percent } +\newlabel{eq: ineq}{{1}{1}{Exercise 35}{equation.1}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Exercise 36}{2}{subsection.1.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Exercise 37}{2}{subsection.1.3}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Exercise 38}{3}{subsection.1.4}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.5}Exercise 39}{3}{subsection.1.5}\protected@file@percent } +\newlabel{eq: smf}{{30}{3}{Exercise 39}{equation.30}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.6}Exercise 40}{3}{subsection.1.6}\protected@file@percent } +\newlabel{eq: opp}{{38}{3}{Exercise 40}{equation.38}{}} +\gdef 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[2][-]{subsection.1.5}{\376\377\000E\000x\000e\000r\000c\000i\000s\000e\000\040\0003\0009}{section.1}% 6 +\BOOKMARK [2][-]{subsection.1.6}{\376\377\000E\000x\000e\000r\000c\000i\000s\000e\000\040\0004\0000}{section.1}% 7 diff --git a/nlin_opt/build/sesh6.pdf b/nlin_opt/build/sesh6.pdf Binary files differ. diff --git a/nlin_opt/build/sesh6.toc b/nlin_opt/build/sesh6.toc @@ -0,0 +1,7 @@ +\contentsline {section}{\numberline {1}Sheet 6}{1}{section.1}% +\contentsline {subsection}{\numberline {1.1}Exercise 35}{1}{subsection.1.1}% +\contentsline {subsection}{\numberline {1.2}Exercise 36}{2}{subsection.1.2}% +\contentsline {subsection}{\numberline {1.3}Exercise 37}{2}{subsection.1.3}% +\contentsline {subsection}{\numberline {1.4}Exercise 38}{3}{subsection.1.4}% +\contentsline {subsection}{\numberline {1.5}Exercise 39}{3}{subsection.1.5}% +\contentsline {subsection}{\numberline {1.6}Exercise 40}{3}{subsection.1.6}% diff --git a/nlin_opt/preamble.tex b/nlin_opt/preamble.tex @@ -51,8 +51,8 @@ \markright{Popović\hfill Nonlinear Optimization\hfill} - \title{University of Vienna\\ Faculty of Mathematics\\ \vspace{1cm}Nonlinear Optimization Problems } +\date{} \author{Milutin Popovic} diff --git a/nlin_opt/sesh1.tex b/nlin_opt/sesh1.tex @@ -0,0 +1,329 @@ +\include{./preamble.tex} + +\begin{document} +\maketitle +\tableofcontents +\section{Sheet 1} +\subsection{Exercise 1} +Let $X \subseteq \mathbb{R}^{n}$ be an nonempty convex set and +$f:\mathbb{R}^{n} \to \mathbb{R}$ a convex function. Show that every local +minimum of $f$ w.r.t. $X$ is a global minimum of $f$ w.r.t. $X$. +\newline + +Let $x^{*} \in X$ be the local minimum of $f$ w.r.t. $X$. Then there is an +$\varepsilon$-ball $B_{\varepsilon}(x^{*}) = \{x \in X:\ \|x^{*}- x\|\le +\varepsilon\}$ such that $f(x^{*}) \le f(x)$ for all $x \in +B_\varepsilon(x^{*}) \cap X$. Now suppose $x^{*}$ is not a global minimum +then there is a $x_{0} \in X$ such that $f(x_{0}) < f(x^{*})$. Since $X$ is +convex there is a line connecting $x_{0}$ and $x^{*}$ in $X$. This line has +elements of the form +\begin{align} + L = \left\{\left( 1-\lambda \right)x_0 - \lambda x^{*}:\ \lambda \in[0,1] + \right\} + \subseteq X +\end{align} +Now for all $z \in L$ with $z = (1-\lambda)x_0 - \lambda x^{*}$ ($\forall +\lambda \in [0, 1]$) it holds that +\begin{align} + &\|x^{*}- (1-\lambda)x_0 - \lambda x^{*}\| \le \varepsilon \\ + &(1-\lambda) \|x^{*}- x_{0}\| \le \varepsilon \; \Rightarrow \; \lambda + \simeq 1 +\end{align} +and +\begin{align} + f\left( \left( 1-\lambda \right)x_{0} +\lambda x^{} \right) + &\le \lambda f(x^{*}) + (1-\lambda) f(x_0)\\ + &< \lambda f(x_0) + (1-\lambda)f(x_0)\\ + &= f(x_0) +\end{align} +This means that we found a $\tilde{x} = \lambda x^{*} - (1-\lambda)x_0 \in +B_\varepsilon(x^{*})$ such that $f(\tilde{x}) < f(x^{*})$ which is a +contradiction since $x^{*}$ is a local minimum in $B_\varepsilon(x^{*})$. +This means that $x^{*}$ is a global minimum of $f$ w.r.t $X$. +\subsection{Exercise 2} +Let $X \subseteq \mathbb{R}^{n}$ be a nonempty, $x_0 \in X$. Show that +\begin{enumerate} + \item $T_X(x_0)$ is a nonempty closed cone + \item If $X$ is convex, then $T_X(x_0) = \text{cl}\left( \bigcup_{\lambda + \ge 0 } \lambda \left( X - x_0 \right) \right)$ + \item If $X$ is convex then $\left( T_X(x_0))^{*} = - N_X(x_0) \right) $ +\end{enumerate} +The statements will be proven in chronological order. Starting with $1.$ the +Bouligand tangent cone to $X$ is defined as +\begin{align} + T_X(x_0) = \{d \in \mathbb{R}^{n}: \exists (x^{k})_{k\ge 0}\subset X, \exists + (t_k)_{k\ge 0}\searrow 0 :\ \frac{x^{k} -x_0}{t_k}\to d\} +\end{align} +Now $T_X(x_0)$ is nonempty since for $x^{k} = x_0$ for all $k$ and a sequence $t_k = +\frac{1}{k}$ we have +\begin{align} + \frac{x^{k}-x_0}{t_k} \to 0 \in T_X(x_0) +\end{align} +To show that $T_X(x_0)$ is closed, consider a sequence $(d_k)_{k\ge 0}\subset +T_X(x_0)$ with convergence point $d \in \mathbb{R}^{n}$. To show that it is closed we +need to show that $d \in T_X(x_0)$. So for all $d_n \in (d_k)_{k\ge 0} +\subset T_X(x_0)$ there exists a sequence $(x^{n,k})_{k\ge 0} \subset X$ and +a sequence $(t_{n, k})_{k\ge 0}$ with $t_{n, k} \searrow 0$ as $k \to +\infty$ such that +\begin{align} + &\frac{x^{n,k}-x_0}{t_{n,k}} \to d_n \;\;\; \forall n\ge 0,\\ + &\text{and}\nonumber\\ + &\lim_{n\to \infty}\lim_{k \to \infty}\frac{x^{n,k}-x_0}{t_{n,k}} = d. +\end{align} +Then there exist $K_\varepsilon$ and $N_\delta$ for $\varepsilon,\delta >0$ +such that +\begin{align} + &\|d_k - d\|\le \varepsilon \;\; \forall k\ge K_\varepsilon,\\ + &\|\frac{x^{n,K_\varepsilon}-x_0}{t_{n,K_\varepsilon}} - + d_{K_\varepsilon}\| \le \delta \;\; \forall n \ge N_\delta. +\end{align} +To conclude the proof consider +\begin{align} + &\|\frac{x^{N_\delta,K_\varepsilon}-x_0}{t_{N_\delta,K_\varepsilon}} - + d\|\\ + &= \|\left(\frac{x^{N_\delta,K_\varepsilon}-x_0}{t_{N_\delta,K_\varepsilon}} - + d_{K_\varepsilon}\right) + \left( d_{K_\varepsilon} - d\right) \|\\ + &\le \varepsilon + \delta +\end{align} +Meaning that $d \in T_X(x_0)$. +Then +$T_X(x_0)$ is really a cone because $\forall d \in T_X(x_0)$ and $\lambda +>0$ we have that $\lambda d \in T_X(x_0)$ by the choice $t_{k,\lambda}= +\frac{1}{\lambda}t_k$ +\begin{align} + \frac{x^{k}- x}{t_{k,\lambda}} = \lambda \frac{x^{k}-x_0}{t_k}\to + \lambda d \in T_X(x_0) +\end{align} +For number 2 additionally $X$ is a convex set. And by +definition of a cone $\bigcup_{\lambda \ge 0}(X-x_0)$ is a cone. And the +union of convex sets is also convex the set +$\text{cl}\left(\bigcup_{\lambda\ge_0} \lambda(X-x_0 ) \right)$ is also. +convex. +For number 3 we simply calculate +\begin{align} + -\left( T_X(x_0) \right)^{*} + &= -\{s \in R^{n}: s^{T}d \ge 0 \forall d \in T_X(x_0)\}\\ + &= \{s \in R^{n}: s^{T}d \le 0 \forall d \in T_X(x_0)\}\\ + &= \{s \in R^{n}: s^{T}(x-x_0) \le 0 \forall x \in X\} +\end{align} +Since $forall x \in X$ there exists an appropriate sequence converging to $x$ +subjected to the tangent cone elements. +\subsection{Exercise 3} +Let $X \subseteq \mathbb{R}^{n}$ be nonempty and $\text{dist}_X: \mathbb{R}^{n} +\to \mathbb{R}$, with $\text{dist}_X(y) = \inf \{\|y-x\|: x\in X\}$. Then consider the +directional derivative of a function $f: \mathbb{R}^{n}\to \mathbb{R}$ at +direction $d$ at $x_0 \in \mathbb{R}^{n}$ +\begin{align} + f'(x_0; d) = \lim_{t \downarrow 0} \frac{f(x_0 + t d) - f(x_0)}{t} +\end{align} +Show that if $X \subseteq \mathbb{R}^{n}$ is nonempty and convex then the +tangent cone can be written as +\begin{align} + T_X(x_0) = \{d \in \mathbb{R}^{n}:\; \left( \ \text{dis}t_X + \right)'(x_0;d) = 0\}. +\end{align} +First we note +\begin{align} + (\text{dist}_X)'(x_0;d) = \lim_{t \downarrow 0} \frac{\text{dist}_X(x_0+t + d)}{t} = 0, +\end{align} +is true for all $x_0 + t d \in X$. Since $X$ is convex we have that +\begin{align} + T_X(x_0) &= \text{cl}\left( \bigcup_{\lambda}\left( X-x_0 \right) \right) + \\ + &= \text{cl}\left( \{\lambda (x - x_0): x\in X, \lambda \ge 0\} \right) +\end{align} +Then $(\text{dist}_X)'(x_0;d) = 0$ holds only for vectors of the form +$d = \lambda (x - x_0)$, $\lambda \ge 0$ and $x \in X$. +\subsection{Exercise 4} +We consider the general constrained optimization problem +\begin{align} + \text{min}\quad & f(x),\\ + \text{s.t.}\quad & g_i(x) \le 0, i=0,\ldots,m\nonumber\\ + &h_j(x) = 0, j=0,\ldots,p\nonumber\\ + & x \in \mathbb{R}^{n}\nonumber +\end{align} +for $f, g_i, h_j: \mathbb{R}^{n}\to \mathbb{R}$, $i=0,\ldots,m$ and +$j=0,\ldots,p$ continuously differentiable. Let $x_0$ be a feasible element +of the problem and +\begin{align} + X_{\text{lin}} := \{ x \in \mathbb{R}^{n}:\quad + &g_i(x_0) + \nabla g_i(x_0)^{T}(x-x_0), i=0,\ldots,m\\ + &h_j(x_0) + \nabla h_j(x_0)^{T}(x-x_0), j=0,\ldots,p\}. +\end{align} +Show that $T_\text{lin}(x_0) = T_{X_\text{lin}}(x_0)$. +\newline +For the reminder +\begin{align} + T_{\text{lin}}(x_0) := \{ d \in \mathbb{R}^{n}:\quad + &\nabla g_i(x_0)^{T}d, \forall i \in \mathcal{A}(x_0)\\ + &\nabla h_j(x_0)^{T}d, j=0,\ldots,p\}, +\end{align} +where $\mathcal{A}(x_0) = \{i=i,\ldots,m:g_i(x_0) = 0\}$. First we show that +$X_\text{lin}$ is convex then we can use the second part of exercise 2. So +for all $x, y \in X_{\text{lin}}$ we need to show that $z=(1-\lambda)x +\lambda y \in X_{\text{lin}}$, this is done by checking the conditions +\begin{align} + &\nabla g_i(x_0) + \nabla g_i(x_0)^{T} (z-x_0)\\ + &=\nabla g_i(x_0) + \nabla g_i(x_0)^{T} ((1-\lambda)x + \lambda y-x_0)\\ + &=\nabla g_i(x_0) + \nabla g_i(x_0)^{T} (x-\lambda x + \lambda y-x_0)\\ + &=\nabla g_i(x_0) + \nabla g_i(x_0)^{T} (x-x_0) + + \nabla g_i(x_0)^{T}(-\lambda x + \lambda y)\\ + &\le \lambda \nabla g_i(x_0)^{T}(-x + y) \qquad (\lambda \ge 0 )\\ + &\le \nabla g_i(x_0)^{T}(- x + y + x_0 - x_0) + g_i(x_0) + -g_i(x_0)\\ + &=-\nabla g_i(x_0) - \nabla g_i(x_0)^{T} (x-x_0) + + \nabla g_i(x_0) + \nabla g_i(x_0)^{T} (y-x_0)\\ + &\le 0 +\end{align} +and similarly for $h_j$ +\begin{align} + &\nabla h_j(x_0) + \nabla h_j(x_0)^{T} (z-x_0)\\ + &=\nabla h_j(x_0) + \nabla h_j(x_0)^{T} ((1-\lambda)x + \lambda y-x_0)\\ + &=\nabla h_j(x_0) + \nabla h_j(x_0)^{T} (x-\lambda x + \lambda y-x_0)\\ + &=\nabla h_j(x_0) + \nabla h_j(x_0)^{T} (x-x_0) + + \nabla h_j(x_0)^{T}(-\lambda x + \lambda y)\\ + &= \lambda \nabla h_j(x_0)^{T}(-x + y)\\ + &= \lambda \nabla h_j(x_0)^{T}(- x + y + x_0 - x_0) + \lambda h_j(x_0) + -\lambda h_j(x_0)\\ + &=\lambda\left(-\nabla h_j(x_0) - \nabla h_j(x_0)^{T} (x-x_0) + + \nabla h_j(x_0) + \nabla h_j(x_0)^{T} (y-x_0)\right)\\ + &= 0 +\end{align} +Thereby $X_{lin}$ is convex and +\begin{align} + T_{X_\text{lin}}(x_0) = \bigcup_{\lambda \ge 0 }\lambda \left( + X_\text{lin} - x_0 \right) . +\end{align} +Additionally $T_{lin}(x_0)$ is a polyhedral so per definition +\begin{align} + T_\text{lin}(x_0) = \bigcup_{\lambda \ge 0 }\lambda \left( X_\text{lin} - + x_0 \right) +\end{align} +\subsection{Exercise 5} +Let $g: \mathbb{R}^{2} \to \mathbb{R}^{4}$ with +\begin{align} + g(x, y) = + \begin{pmatrix} + \pi - 2x\\ + -y - 1\\ + 2x - 3\pi\\ + y - \sin(x) + \end{pmatrix}. +\end{align} +and +\begin{align} + X = \{(x,y) \in \mathbb{R}^{2}: g(x,y) \leqq 0\}. +\end{align} +The set $X$ has the following constrains on $x, y$ +\begin{align} + \pi - 2x \le 0 \; \Rightarrow x \ge \frac{\pi}{2}\\ + -y - 1 \le 0 \Rightarrow y \ge -1\\ + 2x - 3\pi \le \Rightarrow x \le \frac{3\pi}{2}\\ + y - \sin\left( x \right) \le 0 \Rightarrow y \le 1, +\end{align} +meaning $x \in [\frac{\pi}{2}, \frac{3\pi}{2}]$ and $y \in [-1, 1]$. +The tangent cones of $X$ w.r.t $x_1 = (\frac{\pi}{2}, 1)^{T}, +x_2=(\pi,0)^{T}$ and $x_3 = (\frac{3\pi}{2}, -1)^{T}$ are +\begin{align} + T_X(x_1) = \{(0, -\lambda)^{T}, (\lambda, 1)^{T}: \lambda > 0\} \\ + T_X(x_2) = \{\lambda(\cos(x), \sin(x))^{T}, (\lambda, 1)^{T}: \lambda > 0\} \\ + T_X(x_3) = \{(-\lambda, 0)^{T}, (0, \lambda)^{T}: \lambda > 0\} +\end{align} +Graphically it represents the following +\begin{figure}[H] + \centering + \includegraphics[width=0.4\textwidth]{./build/pics/sesh1_ex5.png} + \caption{Graphical representation of $X$ and tangent cones $T_X(x_i)$, $i=1, 2, + 3$} + \label{fig:} +\end{figure} +Then we look at the linearized tangent cones $T_\text{lin}(x_i) = \{d\in \mathbb{R}^{2}: +\nabla g_i(x_i)^{T} d \le 0 \forall i \in \mathcal{A}(x_i)\}$. First we +calculate the gradients of entries of $g$ +\begin{align} + \nabla g_1(x, y) = (-2 ,0)^{T}\qquad \nabla g_2(x,y) = (0, -1)^{T}\\ + \nabla g_3(x, y) = (2 ,0)^{T}\qquad \nabla g_4(x,y) = (-\cos(x), 1)^{T} +\end{align} +then for all $j \in \{ 1, 2, 3, 4\} $ and $i \in \{1, 2, 3\}$ we check if +$g_j(x_i) \le 0$ and construct $\mathcal{A}(x_i)$ and then find $d$ subjected +to the condition. +For $x_1$ we have +\begin{align} + g_1(x_1) = 0,\;\; + g_2(x_1) \neq 0,\;\; + g_3(x_1) \neq 0,\;\; + g_4(x_1) = 0 \;\;\\ + \Rightarrow \mathcal{A}(x_1) = \{1, 2\} +\end{align} +Then +\begin{align} + T_{\text{lin}} + &= \{d \in \mathbb{R}^{2} : (0, 1)d \le 0, (-2, 0)d \le 0\}\\ + &=\{(\lambda, 0)^{T} , (0, -\lambda)^{T}: \lambda >0 \} +\end{align} +For $x_2$ we have +\begin{align} + g_1(x_1) \neq 0,\;\; + g_2(x_1) \neq 0,\;\; + g_3(x_1) \neq 0,\;\; + g_4(x_1) \neq 0 \;\;\\ + \Rightarrow \mathcal{A}(x_1) = \{4\} +\end{align} +Then +\begin{align} + T_{\text{lin}} + &= \{d \in \mathbb{R}^{2} : (1, 1)d \le 0\}\\ + &=\{(-\lambda, \lambda)^{T} , (\lambda, -\lambda)^{T}, (-\lambda, 0)^{T}, (0, + -\lambda)^{T}: \lambda >0 \} +\end{align} +For $x_3$ we have +\begin{align} + g_1(x_1) \neq 0,\;\; + g_2(x_1) = 0,\;\; + g_3(x_1) = 0,\;\; + g_4(x_1) = 0 \;\;\\ + \Rightarrow \mathcal{A}(x_1) = \{2, 3, 4\} +\end{align} +Then +\begin{align} + T_{\text{lin}} + &= \{d \in \mathbb{R}^{2} : (0, -1)d \le 0, (2, 0)d\le 0, (0, 1)d \le 0\}\\ + &=\{(-\lambda, 0)^{T}: \lambda >0 \} +\end{align} +We conclude that $T_{X_\text{lin}}(x_i) = T_{\text{lin}}(x_i)$ only for $i=1$. +\subsection{Exercise 6} +Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^{m}$. Prove using +the strong duality theorem of linear optimization that the following +statements are equivalent +\begin{enumerate} + \item The system $Ax = b$ has a solution $x \leqq 0$ + \item It holds $b^{T}d \ge 0$ for all $d \in \mathbb{R}^{m}$ with $A^{T}d + \leqq 0$ +\end{enumerate} +we show that $1$ is equivalent to $\neg \exists d \in R^{m}: Ad \le 0$ +and $b^{T}d > 0$. Consider the following primal dual +\begin{align} + \text{max}\quad & 0^{T}x\\ + \text{s.t.:}\quad & Ax=b\nonumber\\ + &x\leqq 0\nonumber\\ + \text{and}\nonumber\\ + \text{min}\quad & b^{T}d\\ + \text{s.t.:}\quad & Ad\le 0\nonumber\\ + &d \in R^{m}\nonumber\\ +\end{align} +Consider a solution of $Ax = b$ such that $x \ge 0$. This means that the +primal is true, so there exists an optimal solution since $0^{T}x=0$ then $0$ +must be this primal optimal. By the duality $0$ must be the optimal of the +dual problem, which means $b^{T}d =0 $ so there is no $d \in \mathbb{R}^{m}: +b^{T}d >0$.\newline +On the other hand $\neg \exists d \in \mathbb{R}^{m}: Ad \le 0$ and $b^{T}d +>0$ so $\forall d' \in \mathbb{R}^{m}$ we have that $Ad' > 0$ or $b^{T}d \le +0$ such that $b^{T}d \leq 0 $ for all $d \in \mathbb{R}^{m}$. So we can +conclude that there exists a solution because the primal has at least one +feasible $x \in \mathbb{R}^{m}: Ax = 0, x\geqq 0$. Thereby the two statements +above are equivalent. + +\end{document} + +
