From 7a5312a468b867bb85e039302e4307ea4a277c1d Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Wed, 9 Oct 2024 23:00:22 +0800 Subject: [PATCH] update examples --- LowdimLP.pdf | Bin 537820 -> 544975 bytes LowdimLP.tex | 37 ++++++++++++++++++++++++++++--------- 2 files changed, 28 insertions(+), 9 deletions(-) diff --git a/LowdimLP.pdf b/LowdimLP.pdf index 465ff8febdd07467b5d4765ca638fd0c090f5602..62af268021d8c38b8d6629c9dfe151adc30f5be2 100644 GIT binary patch delta 110917 zcmZs?Q*>ov*R5MCR>iKMV%xTD+qSV{TPs$@NyVtxwr#s&C#SxD?|-YE*5<|A+q|9g z8E+qB^o`S>(*;BcqyScs6ABD%Tg6373F@pl|xd44f=!I91CdNmg`ZIG*OHs859q1Xh-=0+o`tw zG!er{LIbERNhm4u;e>3z!Bb(oP!jI4b9wyEYU5aDw;Ix>h%V*ElXdRQ9unb*oG7lW zd>x$Bc+apoe@kIHoo~NIkGWMKF|`!!#j_xEyL9=$9hWF@T~&u)Zn~Sf0a>yOYUBSPns#&gJ;NN~;{D`zp^2&-^F=Xb;&ZdEJ0f}EA@VLc;RGc;7$i1`NF zVSj)5puZ3>SOCNX2&unfyf8!7hq7SYi&2fCko(Xq6 ziTphG_j`2yHh@@g;ltVaw15MpL^&UTlo%6Bcyfuq=7HId!68S_g+N$^I~~&MC^Mjv z#-;XAKfPAm5G}mXcJgeVX)Xy4^+c9g;+L4k-If>cDI{fuexMURY^>TfFWp#&^_L|4 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and 3 breakpoints} \label{fig:sub1} \end{subfigure}% \begin{subfigure}{.5\textwidth} \centering \includegraphics[width=.4\linewidth]{images/Piecewise_linear_function2D.png} - \caption{A 2D piecewise concave function} + \caption{A 2D pwl concave function} \label{fig:sub2} \end{subfigure} % \caption{A figure with two subfigures} @@ -93,9 +92,11 @@ However, observe that in our problem the piecewise linear convex function is not \end{frame} \section{LP in Low Dimensions} -\begin{frame}[allowframebreaks]{Megiddo's algorithm} - {\tiny \url{https://people.inf.ethz.ch/gaertner/subdir/texts/own_work/chap50-fin.pdf}} - +\begin{frame}[allowframebreaks]{Megiddo's algorithm}% + \mybox[oliver!20]{ + \tiny + \url{https://people.inf.ethz.ch/gaertner/subdir/texts/own_work/chap50-fin.pdf} + } The dimension $d$ (in our problem, the dimension of $x$) is small while the number of constraints are huge. We need only $d$ linearly independent tight constraints to identify the optimal solution $x^*$. Thus most of the constraints are useless. \BlankLine @@ -167,13 +168,14 @@ However, observe that in our problem the piecewise linear convex function is not with solution $T(n,d)=O(2^{2^d}n)$. \end{frame} \begin{frame}{Zemel's conversion} - Our linear program has \emph{dimension} $n+d$. \href{https://www.sciencedirect.com/science/article/abs/pii/0020019084900140}{Zemel} showed that this kind of problem can be converted to a linear program of dimension $d$. \begin{align*} \min &\sum_{i=1}^n f_i\\ s.t. \quad f_i&\geq \alpha_j(a_i\cdot x -b_i)-\beta_j \quad \forall i\in[n], \forall j\\ \end{align*} + Our linear program has \emph{dimension} $n+d$. + \textbf{\href{https://www.sciencedirect.com/science/article/abs/pii/0020019084900140}{Zemel}} showed that this kind of problem can be solved in linear time. \mybox[oliver!20]{ - \scriptsize Here is an intuitive way to understand the conversion. One can think the LP above as a $d$-dimensional search problem with $n+d$ hyperplanes. However, the oracle is quite different. The oracle takes the unknown $x^*$ and a hyperplane $H$ as input, returns the relative position by computing the minimal $f_i$. + This is a \emph{$d$-dimensional search problem} with $n+d$ hyperplanes. } \end{frame} @@ -205,15 +207,32 @@ However, observe that in our problem the piecewise linear convex function is not If a LP problem has low dimension, then its combinatorial dimension is low. \textbf{What about the converse?} \newpage + \begin{align*} + \min &\sum_{i=1}^n f_i\\ + s.t. \quad f_i&\geq \alpha_j(a_i\cdot x -b_i)-\beta_j \quad \forall i\in[n], \forall j\\ + &\cdots + \end{align*}% \textbf{Does our LP has low combinatorial dimension?} - No! A basis contains at least $n$ constraints since otherwise some $f_i$ is unbounded. + No. A basis contains at least $n$ constraints since otherwise some $f_i$ is unbounded. + + \begin{problem} + Is it possible to formulate the pwl convex minimization problem as an LP-type problem with low combinatorial dimension? + \end{problem} \end{frame} \begin{frame}{Aggregate the pwl convex functions} % blog posts + The sum of pwl convex functions are still pwl convex. + \newline If we can compute $F=\sum f_i$ in $O(m)$ and the number of line segments on $F$ is also $O(m)$, then the corresponding LP will have low combinatorial dimension. + \begin{align*} + \min \quad F\\ + s.t. \quad F&\geq \alpha_j\cdot x -\beta_j \quad \forall j\\ + &\cdots + \end{align*} + However, this is not possible for general pwl convex functions in $\R^d$.\footnote{see this \href{https://talldoor.uk/posts/2024-09-16-piecewise-linear.html}{blog post} for detail.} \end{frame} \end{document} \ No newline at end of file