\mathsurround 1pt
 
%       brackets, parenthesis, and other miscelaneous marks
\newcommand{\lp}{\left\{}
\newcommand{\rp}{\right\}}
\newcommand{\lb}{\left(}
\newcommand{\rb}{\right)}
\newcommand{\ls}{\left|}
\newcommand{\rs}{\right|}
\newcommand{\Rbrack}{\right]\!\right]}
\newcommand{\Lbrack}{\left[\!\left[}
\newcommand{\eqdef}{\stackrel{\rm def}{=}}
\newcommand{\nn}{\nonumber}
\newcommand{\D}{\partial}
 
%       graph notation
\newcommand{\Wcomp}{\V\backslash\W}
\newcommand{\cutVW}{\Lbrack\V,\Wcomp\Rbrack}
\newcommand{\cutWV}{\Lbrack\Wcomp,\V\Rbrack}
\newcommand{\graph}{\lb\V,\E\rb}
\newcommand{\wbargraph}{\lb\V,\E,\abar\rb}
\newcommand{\wgraph}{\lb\V,\E,a\rb}
\newcommand{\graphA}{\lb\V,\E_A\rb}
\newcommand{\Graph}[1]{\lb\V^#1,\E^#1,a^#1\rb}
\newcommand{\Gcon}[1]{\G/#1}
\newcommand{\mean}[1]{\frac{1}{\ls#1\rs}\sum_{e\in#1} a_e}
 
%       sets and functions
\newcommand{\Rnn}{\Re_{n\times n}^+}
\newcommand{\Rn}{\Re_n^+}
\newcommand{\Am}{\{A_m\}}
\newcommand{\dphi}{\phi^\prime}
% UVC: \newcommand{\grad}{\bigtriangledown}
\newcommand{\cost}{C\!\left(\cdot\right)}
\def\QED{\hfil\vrule height 5pt width 4pt depth 0pt} 
%       vectors and such
\newcommand{\1}{{\bf1}}
 
%       script\Greek letters
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\R}{\mathcal{R}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\Cs}{\mathcal{C}_{\bf s}}
\newcommand{\bs}{{\bf s}}
\newcommand{\DD}{\mathcal{D}}
\newcommand{\bsg}{{\bf s}_\gamma}
\newcommand{\Qs}{Q_{\bf s}}
\newcommand{\Ws}{{\bf W}_{\bf s}}
\newcommand{\At}{A^T}
\newcommand{\Gg}{G_\gamma}
\newcommand{\E}{\mathcal{E}}
\newcommand{\G}{\mathcal{G}}
\newcommand{\Hy}{\mathcal{H}}
\newcommand{\I}{\mathcal{I}}
\newcommand{\K}{\mathcal{K}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\pr}{{\prime}}
\newcommand{\Pa}{\mathcal{P}}
\newcommand{\Ss}{\mathcal{S}}
\newcommand{\T}{\mathcal{T}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\V}{\mathcal{V}}
\newcommand{\W}{\mathcal{W}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\Y}{\mathcal{Y}}
\newcommand{\Z}{\mathcal{Z}}
\newcommand{\g}{{\gamma}}
\newcommand{\de}{{\delta}}
\newcommand{\lam}{{\lambda}}
\newcommand{\gbar}{{\bar\gamma}}
 
%        bars and hats
\newcommand{\Dbar}{{\bar D}}
\newcommand{\abar}{{\bar a}}
\newcommand{\bbar}{{\bar b}}
\newcommand{\mbar}{{\bar m}}
\newcommand{\nbar}{{\bar n}}
\newcommand{\vbar}{{\bar v}}
\newcommand{\wbar}{{\bar w}}
\newcommand{\xbar}{{\bar x}}
\newcommand{\ybar}{{\bar y}}
\newcommand{\zbar}{{\bar z}}
\newcommand{\chat}{{\hat c}}
\newcommand{\rhat}{{\hat r}}
\newcommand{\delhat}{{\hat \delta}}
 
%       relations and implications
\newcommand{\ra}{\rightarrow}
% \newcommand{\implies}{\Longrightarrow}
% \newcommand{\impliedby}{\Longleftarrow}
\newcommand{\qimpliesq}{\quad\implies\quad}
\newcommand{\qiffq}{\quad\iff\quad}
\newcommand{\qandq}{\quad\mbox{and}\quad}
\newcommand{\qotherwise}{\quad\mbox{otherwise}}
\newcommand{\qand}{\quad\mbox{and}}
\newcommand{\qif}{\quad\mbox{if }}
\newcommand{\qfor}{\quad\mbox{for }}
\newcommand{\qforevery}{\quad\mbox{\rm for every }}
\newcommand{\qforany}{\quad\mbox{\rm for any }}
 
 
%       summations, mins and maxes, products, and limits
\newcommand{\sumC}{\sum_{(i,j)\in \C}}
\newcommand{\sumdelm}{\sum_{(i,j)\in \delta^-}}
\newcommand{\sumdelp}{\sum_{(j,i)\in \delta^+}}
\newcommand{\sumg}{\sum_{e\in\gamma}}
\newcommand{\sumgstar}{\sum_{e\in\gamma^*}}
\newcommand{\sumG}{\sum_{\g\in\Gamma}}
\newcommand{\sumvplus}{\sum_{e\in\delta^+_v}}
\newcommand{\sumvminus}{\sum_{e\in\delta^-_v}}
\newcommand{\maxG}{\max_{\gamma\in\Gamma}}
\newcommand{\minG}{\min_{\gamma\in\Gamma}}
\newcommand{\maxE}{\max_{e\in\E}}
\newcommand{\minE}{\min_{e\in\E}}
\newcommand{\minV}{\min_{v\in\V}}
\newcommand{\maxV}{\max_{v\in\V}}
\newcommand{\maxvplus}{\max_{e\in\delta^+_v}}
\newcommand{\maxvminus}{\max_{e\in\delta^-_v}}
\newcommand{\maxk}{\max_{0\le k\le n-1}}
\newcommand{\prodg}{\prod_{e\in\g}}
\newcommand{\limf}{\lim_{n\to\infty}}
\newcommand{\limfb}{\lim_{\bar n\to\infty}}
\newcommand{\imin}[4]{\min_{#1\le#2\le#3}\lp #4 \rp}
\newcommand{\ahalf}{\frac{1}{2}} 
\newcommand{\vareps}{\varepsilon}                                                                    
\newcommand{\noref}{\nocite}
%       theorems, and such
\newtheorem{theom}{Theorem}
%\newtheorem{conj}{Conjecture}
\newtheorem{lemma}{Lemma}
%\newtheorem{prop}{Proposition}
%\newtheorem{cor}{Corollary}
\newtheorem{defin}{Definition}
%\newtheorem{prob}{Problem}
%\newtheorem{assum}{Assumption}
%\newtheorem{ex}{Example}
%\newtheorem{prf}{Proof}


 
%       math program, etc
\newenvironment{mathprog}{\[ \begin{tabular}[t]{rl}}{\end{tabular}\end{tabular} \]}
\newcommand{\obj}[3]{\raisebox{-1.5ex}{$\stackrel{\mbox{#1}}{#2}$} & $\displaystyle#3$ \\ \vspace{\bigskipamount}}
\newcommand{\st}{subject to & \begin{tabular}[t]{rcl}}
\newcommand{\con}[3]{$\displaystyle#1$ & $\displaystyle\!\!\!\! #2 \!\!\!\!$ & $\displaystyle#3$ \\ }






















