%  current problems to resolve:
% two different defs.   \newcommand{\cR}{I\!\! R} %   \def\cR{{\cal R}}
%%%   first definition is now used; cal R is commented out!!
%%%
%    what is   \ex ????   see  \nc below
%%%%%%%%%%%%%%%%%%%55
%   environments were changed, e.g.
%         \newtheorem{Theorem}{Theorem}[section]   from
%         \newtheorem{Theorem}[Definition]{Theorem}
%   This will change some numbering that authors may have expected.
%         
%
%
%
% 
%     Pagelayout and packages used
%\oddsidemargin=.2cm
%\textwidth=6.5in
%\topmargin=-.5in
%\textheight=8.5in
%\textheight22cm
%\input psfig
%\usepackage{amsmath}
%\usepackage{amssymb}   %%%???why trouble???
\usepackage{accents}
\usepackage{subeqn}    %  from Renato
\usepackage{amsfonts}
\usepackage{psfrag,lscape,graphicx}
%\usepackage[mathcal,mathbf]{euler}
%\usepackage{theorem,amsmath,enumerate,fancyhdr,amssymb,accents,amsfonts}
%\usepackage{showkeys}
%\usepackage{latexsym}
\input{epsf}
%\usepackage{epsfig}
%\oddsidemargin -12pt
%\evensidemargin -12pt
%\topmargin -27pt
%\textheight 635pt
%\textwidth 460pt
%\def\baselinestretch{1.2}

%     New environments
%\newtheorem{subequations}{Subequations}[section] 
                    %overridden by style file subeqn from Renato
\newtheorem{thm}{Theorem}[section]
\newtheorem{theorem}{Theorem}[section]  %changed from edbk.cls file
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{valtheorem}{Theorem}[section]
%\newtheorem{algorithm}{Algorithm}[section]
\newtheorem{alg}{Algorithm}[section]
\newtheorem{assumption}{Assumption}[section]
\newtheorem{axiom}{Axiom}[section]
\newtheorem{condition}{Condition}[section]
\newtheorem{conj}{Conjecture}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{Corollary}{Corollary}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{Def}{Definition}[section]
\newtheorem{defi}{Definition}[section]
\newtheorem{Definition}{Definition}[section]
\newtheorem{defn}{Definition}[section]
\newtheorem{ex}{Exercise}[section]
\newtheorem{exercise}{Exercise}[section]
\newtheorem{example}{Example}[section]
\newtheorem{examp}{Example}[section]
\newtheorem{Example}{Example}[section]
\newtheorem{Fact}{Fact}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{Lemma}{Lemma}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{note}{Note}[section]
\newtheorem{openproblem}{Open problem}[section]
\newtheorem{property}{Property}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{Proposition}{Proposition}[section]
\newtheorem{rem}{Remark}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{Remark}{Remark}[section]
\newtheorem{wff}{Wff}[section]

%\newtheorem{Proposition}[Definition]{Proposition}
%\newtheorem{Theorem}[Definition]{Theorem}
%\newtheorem{Example}[Definition]{Example}
%\newtheorem{Corollary}[Definition]{Corollary}
%\newtheorem{Remark}[Definition]{Remark}
%\newtheorem{Definition}{Definition}[section]
%\newtheorem{Lemma}[Definition]{Lemma}

%     New commands
\newcommand{\nc}{\newcommand}
\nc{\beq}{ \begin{equation} }
\nc{\eeq}{ \end{equation} }
\nc{\bt}{ \begin{tabular} }
\nc{\et}{ \end{tabular} }
\nc{\kwqqp}{Q{$^2$}P\,}
\nc{\kwqqps}{Q{$^2$}Ps}
\nc{\req}[1]{(\ref{#1})}
\nc{\be}{\begin{equation}}
\nc{\ee}{\end{equation}}
\nc{\1}{\;\;\; }

\nc{\vgap}{\vspace*{.1in}}
\nc{\ba}{\begin{array}}
\nc{\ea}{\end{array}}
\nc{\bea}{\begin{eqnarray}}
\nc{\eea}{\end{eqnarray}}
\nc{\bean}{\begin{eqnarray*}}
\nc{\eean}{\end{eqnarray*}}
\nc{\bu}{\bullet}
\nc{\nn}{\nonumber}
\nc{\kron}{\otimes}
\nc{\skron}{\kron_s}
\nc{\cond}{\mbox{\bf cond}}
\nc{\lmin}{\lambda_{\min}}
\nc{\lmax}{\lambda_{\max}}
\nc{\Snn}{{\cal S}^{n^2}}
\nc{\Snnp}{{\cal S}^{n^2}_+}
\nc{\Snnpp}{{\cal S}^{n^2}_{++}}
\nc{\vv}{\mbox{\bf vec\hspace{.03cm}}}
\nc{\tmu}{\widetilde \mu}
\nc{\bra}{\ll}
\nc{\ket}{\gg}
%\nc{\cE}{{\rm I\hspace{-0.7mm}E}} % from Shapiro but not used
\nc{\cp}{\mbox{{\tiny$\,^\bullet$}} }
\nc{\mi}{\mbox{\scriptsize min}}

\nc{\cR}{I\!\! R}
\nc{\cN}{I\!\! N}
\nc{\sect}{\section}

\nc{\eqref}[1]{{\rm (\ref{#1})}}
\nc{\Mmn}{{\cal M}_{m,n} }
\nc{\Kprod}{\otimes}
\nc{\Lam}{\Lambda}
\nc{\Sbar}{\bar{S}}
\nc{\Tbar}{\bar{T}}
\nc{\set}[1]{\{ #1 \}}
\nc{\pibar}{\bar{\pi}}


%   are newcounter commands needed????  - commented out for now
%\newcounter{rulenum}
%\newcounter{sent}
%\newcounter{tempcnt}



\nc{\bee}{\begin{equation}}
\nc{\ene}{\end{equation}}
\nc{\beea}{\begin{eqnarray}}
\nc{\enea}{\end{eqnarray}}
\nc{\minim}{\mathop{\hbox{\rm minimize}}}
\nc{\subsetneqq}{\lower 1.0 pt\hbox{\ $\scriptstyle\ne\llap{\raise 5.7pt\hbox{%
                $\scriptstyle\subset$}}$\ }}
\nc{\minimize[1]}{{\displaystyle\minim_{#1}}} 
\nc{\sol}{\mathop{\rm Sol}}
\nc{\pr}{\mbox{\rm Pr}}
\nc{\sgn}{\mbox{\rm sign}}
\nc{\QED}{\hfill ~\rule[-1pt] {8pt}{8pt}\par\medskip ~~}
%%change to Kluwers QED  ???
%\def\endproof{\ifqedequation\global\qedequationfalse\vskip-3pt%
%\vskip-\belowdisplayskip\hfill\slug\else
%\ifinmathqed\else\slug\vskip1pt\fi\fi%
%\global\inmathqedfalse}
%\nc{\QED}{\endproof}
\nc{\epr}{\QED}
\nc{\qed}{\QED}
\nc{\fb}{\QED}
\nc{\blacksquare}{\QED}
%\def\blacksquare{\hbox{\vrule width 4pt height 4pt depth 0pt}}
%\nc{\fb}{\rule[-2pt]{4pt}{8pt}}
\def\ex{\mbox{\rm E}}    %   this must be used by Kluwer????
          %    could not use     nc   here!!!

%       brackets, parenthesis, and other miscelaneous marks
\nc{\mr}[1]{(\ref{#1})}

%       letters and fractions
\nc{\half}{{\scriptstyle {1 \over 2}}}
\nc{\halff}{ \mbox{\small$\frac{1}{2}$}}
%\newcommand{\half}{\frac{\scriptstyle 1}{\scriptstyle 2}}
%\def\half{\frac{1}{2}}
%\newcommand{\half}{\mbox{${1 \over 2}$}}
%\def\half  {{\textstyle{1\over 2}}}

%       relations and implications
\nc{\rar}{\rightarrow}

% Operations
\nc{\trace}{{\rm Trace\,}}
\nc{\tr}{{\rm Trace\,}}
\newcommand{\rank}{{\rm rank\,}}
\nc{\conv}{{\rm Conv\,}}
%\def\trace{{\rm Trace \,}}
%\def\trace{{\rm	Trace}}
%\def\trace{\mathop{\hbox{\rm trace}}}
%\def\tr{\mathop{\hbox{\rm trace}}}
%\def\tr{{\rm trace \,}}
%\newcommand{\tr}{{\rm trace }}
%\newcommand{\tr}{{\rm tr}}
%\def\tr {\mbox{trace}}
%\newcommand{\trace}{{\rm trace\,}}
%\def\conv{ {\rm Conv \,}}
%\def\conv{\mathop{{\rm Conv}}}
%\def\conv{\mathop{\rm conv}}
%\def\conv{\mathop{{\rm Conv}}}
%\newcommand{\conv}{{\rm conv \afterlog}}

% Sets
\nc{\re}{{\rm I\!R}}
%\def\re{{\rm \Re \,}}
\nc{\rn}{\re^n}
%\newcommand{\rn}{\Re^n}
\nc{\Me}{{\rm I\!M}}
\nc{\Mn}{\Me^{m\times n}}
\nc{\sm}{{\cS^m}}


% Vectors
\nc{\xT}{x^T}
\nc{\xii}{x_i}
\nc{\xk}{x^k}
\nc{\xik}{x_i^k}
\nc{\xTy}{x^Ty}

% Matrices
\nc{\Aij}{A_{ij}}
\nc{\AT}{A^T}
\nc{\Ak}{A^k}
\nc{\AkT}{(A^k)^T}
\nc{\Aik}{A_i^k}
\nc{\AkTij}{((A^k)^T)_{ij}}

% Operators
\nc{\Aop}{{\cal A\/}}
\nc{\Aops}{{\cal A\/}^*}
%\def\Diag{{\mathop{\rm Diag}}}
%\def\Diag{\mathop{{\rm Diag}}}
%\newcommand{\Diag}{{\rm Diag\,}}
%\def\Diag{\mathop{{\rm Diag}}}
%\nc{\Diag}{\mathop{\hbox{\rm Diag}}}
%\nc{\diag}{\mathop{\hbox{\rm diag}}}
%\nc{\Diag}{\rm Diag \smallafterlog}
%nc{\diag}{\rm diag \smallafterlog}
\nc{\diag}{{\rm diag\,}}
\nc{\Diag}{{\rm Diag\,}}
\nc{\adj}{{\rm adj\,}}
\nc{\kvec}{{\rm vec\,}}
\nc{\kmat}{{\rm Mat\,}}
\nc{\arrow}{{\rm arrow\,}}
\nc{\Arrow}{{\rm Arrow\,}}
\nc{\BoDiag}{{\rm B^0Diag\,}}
\nc{\bodiag}{{\rm b^0diag\,}}
%\newcommand{\vec}{{\rm vec\,}}   % already defined as arrow over letter
\newcommand{\svec}{{\rm svec\,}}
\newcommand{\stvec}{{\rm stvec\,}}
\newcommand{\snvec}{{\rm snvec\,}}
\newcommand{\sMat}{{\rm sMat\,}}
\newcommand{\stMat}{{\rm stMat\,}}
\newcommand{\snMat}{{\rm snMat\,}}
\newcommand{\hMat}{{\rm hMat\,}}
\newcommand{\hnMat}{{\rm hnMat\,}}
\newcommand{\htMat}{{\rm htMat\,}}
\newcommand{\dsvec}{{\rm dsvec\,}}
\newcommand{\dstvec}{{\rm dstvec\,}}
\newcommand{\dsnvec}{{\rm dsnvec\,}}
\newcommand{\sdiag}{{\rm sdiag\,}}
\newcommand{\vsMat}{{\rm vsMat\,}}


%Cal commands
\nc{\cA}{{\cal A}}
\nc{\cE}{{\cal E}}
\nc{\cG}{{\cal G}}
\nc{\cF}{{\cal F}}
\nc{\cH}{{\cal H}}
\nc{\Hcal}{{\cal H}}
\nc{\cI}{{\cal I}}
%\nc{\cN}{{\cal N}}
%\nc{\cP}{{\cal P}}
\nc{\cU}{{\cal U}}
%\nc{\Sn}{{\cal S}^n}
\nc{\calS}{{\cal S}}
\nc{\calA}{{\cal A}}
\nc{\cS}{{\cal S}}
\nc{\cP}{{\cal P}}
\nc{\A}{{\cal A}}
\nc{\N}{{\cal N}}
\nc{\E}{{\cal E}}
\nc{\G}{{\cal G}}
\nc{\K}{{\cal K}}
\nc{\F}{{\cal F}}
\nc{\T}{{\cal T}}
\nc{\U}{{\cal U}}
\nc{\Z}{{\cal Z}}
%\newcommand{\M}{{\cal M}}
\nc{\Y}{{\cal Y}}
\nc{\LL}{{\cal L}}
\nc{\I}{{\cal I}}
\nc{\sy}{{\cal S}^n}
\nc{\q}{{\cal Q}}
\nc{\Sn}{{\cal S}^n }
\nc{\Se}{{\cal S}_e }
\nc{\Sd}{{\cal S}_d }
\nc{\Sc}{{\cal S}_C }
\nc{\Sh}{{\cal S}_H }
\nc{\snn}{{\cal S}_{n-1} }
\nc{\KK}{{\cal K} }
\nc{\DD}{{\cal D} }
\nc{\BB}{{\cal B} }
\nc{\PP}{{\cal P} }
\nc{\TT}{{\cal T} }
%\newcommand{\A}{{\cal A} }
\nc{\p}{{\cal P} }



% Miscellaneous
\def\nvec{\mathop{\hbox{\bf nvec}}}
%\def\ostar{\mathop{\ast\hskip-5pt\odot}}
\def\ostar{\odot}
%\def\svec{\mathop{\hbox{\bf svec}}}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
%%% commented out July/99 
\newcommand{\tran}{\mbox{$\top$\hspace{-4.13 mm}
\scriptsize$\cap$}}

\def\Min{\mathop {\rm Min}}
\def\Max{\mathop {\rm Max}}
\def\dis{\mathop{\rm dist}}
\def\val{\mathop{\rm val}}
%\def\cl{\mathop{\rm cl}}
\def\lsc{\mathop{\rm lsc}}
\def\dom{\mathop{\rm dom}}
\def\intt{\mathop{\rm int}}
%% from Monteiro
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
        \hbox{\vrule width.#2pt height#1pt \kern#2pt
        \vrule width.#2pt}
        \hrule height.#2pt}}}}
\def\nr{\par \noindent}
\newcommand{\bpr}{{\bf Proof.} \hspace{1 em}}
\nc{\proof}{\bf Proof. \rm \nr}
%\newcommand{\epr}{ \\ \hspace*{4.5in} $Box$ }  % box needs fixing ????
%\newcommand{\qed}{\hbox{\hspace{1mm}}\hfill $\Box$ \nr}
\def\eqnok#1{(\ref{#1})}
\nc{\dX}{\Delta X}
\nc{\dy}{\Delta y}
\nc{\dS}{\Delta S}
\nc{\tX}{\widetilde X}
\nc{\tE}{\widetilde E}
\nc{\tF}{\widetilde F}
\nc{\tC}{\widetilde C}
\nc{\tA}{\widetilde A}
\nc{\tU}{\widetilde U}
\nc{\tQ}{\widetilde Q}
\nc{\tP}{\widetilde P}
\nc{\tD}{\widetilde D}
\nc{\tb}{\widetilde b}
\nc{\ty}{\widetilde y}
\nc{\tS}{\widetilde S}
\nc{\tdX}{\widetilde {\Delta X}}
\nc{\hdX}{\widehat {\Delta X}}
\nc{\tdS}{\widetilde {\Delta S}}
\nc{\hdS}{\widehat {\Delta S}}
\nc{\tdy}{\widetilde {\Delta y}}
\nc{\hdy}{\widehat {\Delta y}}
\nc{\tV}{\widetilde V}
\nc{\tW}{\widetilde W}
\nc{\hB}{\hat B}
\nc{\hX}{\hat X}
\nc{\hy}{\hat y}
\nc{\hS}{\hat S}
\nc{\hM}{\hat M}
\nc{\hN}{\hat N}
\nc{\hW}{\hat W}
\nc{\hZ}{\hat Z}
\nc{\hC}{\hat C}
\nc{\hA}{\hat {\cal A}}
%\nc{\N}{{\cal N}}
\nc{\tN}{\widetilde{\cal N}}
%%%

\def\la{\langle}
\def\ra{\rangle}
\def\mc{\multicolumn}
\def\norm#1{\|#1\|}
\def\dX{\Delta X}
\def\dy{\Delta y}
\def\dS{\Delta S}
%\def\diag{\mathop{\rm diag}}
%%%\def\R{{\cal R}}
\def\R{I\!\! R}   %%% added and previous commented out.
%\def\R{{\cal I\kern-3.2pt R}}   % changed sl to cal
%\nc{\cS}{S\R}
\def\Rn2{\R^{n(n+1)/2}}
\def\SRn{{\cal S}\R^{n \times n}}
\def\Rn{\R^{n \times n}}
%\def\svec{{\rm \bf svec}}   %  changed to \nc \svec
%\def\vec{{\rm \bf vec}}      %  moved to \nc  \kvec do not use \vec
%\def\smat{{\rm \bf smat}}  % changed to  \nc \sMat
\def\Wq{W^{\frac{1}{2}} }
\def\IWq{W^{-\frac{1}{2}} }
\def\Sq{S^{\frac{1}{2}}}
\def\ISq{S^{-\frac{1}{2}}}
\def\lam{\lambda}
\def\eps{\epsilon}
\def\del{\delta}
\def\kap{\kappa}

\newlength{\len}
\setlength{\len}{5.7in}
\addtolength{\len}{-2\fboxsep}
\addtolength{\len}{-2\fboxrule}

% From Ben-Tal and Nemirovski
\def\huh{\hbox{\vrule width 2pt	height 8pt depth 2pt}}
\def\Bbb#1{{\bf	#1}}
\def\eqnum#1{\eqno (#1)}
%\def\diag{{{\mathop{\rm	diag}}}}   % changed to \nc \diag
%\def\fnote#1{\footnote}  % not used
%\def\blacksquare{\hbox{\vrule width 4pt	height 4pt depth 0pt}}
\def\square{\hbox{\vrule\vbox{\hrule\phantom{o}\hrule}\vrule}}
\def\inter{\mathop{{\rm	int}}}
\def\epi{{\rm epi}}
\def\pn{\par\noindent}
\def\nr{\par \noindent}
\def\diag{{\rm diag}}
\def\compl{{{\mathop{\rm compl}}}}
\def\clsr{{\rm cl}}
%\def\beq{\begin{equation}}
%\def\eeq{\end{equation}}
\def\eqbd{\colon =}
%\def\rhomin{{\mathop{\underline{\rho}}}}
%\def\rhomax{{\mathop{\overline{\rho}}}}
%  changed by Nem. Aug/99
\def\rhomin{\mathop{\underline{\rho}}}
\def\rhomax{\mathop{\overline{\rho}}}
\def\mumax{{\mathop{\mu_{\rm max}}}}
%\newtheorem{exercise}{Exercise}[section]
%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{corollary}{Corollary}[section]
%\newtheorem{proposition}{Proposition}[section]
%\newtheorem{conjecture}{Conjecture}[section]
%\newtheorem{assumption}{Assumption}[section]
%\newtheorem{remark}{Remark}[section]
%\newtheorem{definition}{Definition}[section]
%\newcommand{\proof}{\bf	Proof: \rm \nr}
%\newcommand{\qed}{\hfill $\Box$	\nr \medskip}
%\def\ba{\begin{array}}
%\def\ea{\end{array}}
\def\beann{\begin{eqnarray*}}
\def\eeann{\end{eqnarray*}}
\def\bea{\begin{eqnarray}}
%\def\eea{\end{eqnarray}}
%\def\Tr{\mathop{{\rm Trace}}}
%\def\Det{\mathop{{\rm Det}}}
\def\Det{\mathop{{\rm Det}\,}}
%\def\Sz{\mathop{{\hbox{\sl Size}}}}
%\def\Vl{\mathop{{\hbox{\sl Vol}}}}
%\def\CE{\mathop{{\hbox{\sl CircEll}}}}
%\def\CS{\mathop{{\hbox{\sl CircSimp}}}}
%\def\IE{\mathop{{\hbox{\sl InscrEll}}}}
%%  changed sl above to cal below
\def\Sz{\mathop{{\hbox{\cal Size}}}}
\def\Vl{\mathop{{\hbox{\cal Vol}}}}
\def\CE{\mathop{{\hbox{\cal CircEll}}}}
\def\CS{\mathop{{\hbox{\cal CircSimp}}}}
\def\IE{\mathop{{\hbox{\cal InscrEll}}}}
\def\nrm{\mathop{{\parallel}}}
\def\dim{\mathop{{\rm dim}\,}}
%\def\inter{\mathop{{\rm	int}\,}}
\def\vrt{\mathop{\,\mid}\,}
\def\argmin{\mathop{\rm	argmin}}
%\def\cl{\mathop{{\rm cl}}}
\def\Dom{\mathop{{\rm Dom}}}
\def\Ker{\mathop{{\rm Ker}}}
\def\Compl{\mathop{{\rm	Compl}}}
\def\Accur{\mathop{{\rm	Accur}}}
\def\log{\mathop{{\rm log}}}
\def\mes{\mathop{{\rm mes}}}
\def\Vol{\mathop{{\rm Vol}}}
\def\vol{\mathop{{\rm vol}}}
\def\dist{\mathop{{\rm dist}}}
\def\Argmin{\mathop{\rm	Argmin}}
\def\size{\mathop{{\rm Size}}}
\def\Diam{\mathop{{\rm Diam}}}
\def\AvDiam{\mathop{{\rm AvDiam}}}
\def\Residual{\mathop{{\rm Residual}}}
\def\Size{\mathop{{\rm Size}}}
\def\e{\mathop{{\rm e}}}
\def\sign{\mathop{{\rm sign}}}
\def\EllOut{\mathop{{\rm EllOut}}}
%\def\Ell{\mathop{{\sl Ell}}}   %  cahnged to cal below
\def\Ell{\mathop{{\cal Ell}}}
\def\card{\mathop{{\rm card}}}
\def\TD{{\mathop{\rm TD}}}
\def\minmax{{\mathop{\rm minmax}}}
\def\stable{{\mathop{\rm stable}}}
\def\displ{{\mathop{\rm	displ}}}
\def\book{\cite{NNbook}}
%\def\cR{{\cal R}}
\def\cU{{\cal U}}
\def\cX{{\cal X}}
\def\cF{{\cal F}}
\def\cW{{\cal W}}
\def\bK{{\Bbb K}}
\def\bL{{\Bbb L}}
% From deKlerk et al
\def\m{\multicolumn{2}{l}}
\def\nr{\par \noindent}
%\def\ba{\begin{array}}
%\def\ea{\end{array}}
%\def\beq{\begin{equation}}
%\def\eeq{\end{equation}}
\def\beann{\begin{eqnarray*}}
\def\eeann{\end{eqnarray*}}
\def\bc{\begin{center}}
\def\ec{\end{center}}
%\def\bt{\begin{tabular}}
%\def\et{\end{tabular}}
\def\bmi{\begin{minipage}[t]}
\def\emi{\end{minipage}}
%\def\bea{\begin{eqnarray}}
%\def\eea{\end{eqnarray}}
\def\tz{\tilde{z}}
\def\tu{\tilde{u}}
\def\tp{\tilde{p}}
\def\tpi{\tilde{\pi}}
\def\tq{\tilde{q}}
\def\Tr{{\rm Trace}\,}
\def\tx{\tilde{x}}
\def\ty{\tilde{y}}
\def\ts{\tilde{s}}
\def\tcP{\tilde{\cP}}
\def\tcD{\tilde{\cD}}
\def\tA{\tilde{A}}
\def\tB{\tilde{B}}
\def\tN{\tilde{N}}
\def\tD{\tilde{D}}
\def\tP{\tilde{P}}
\def\tX{\tilde{X}}
\def\tS{\tilde{S}}
\def\tL{\tilde{L}}
\def\tOmega{\tilde{\Omega}}
\def\tb{\tilde{b}}
\def\tc{\tilde{c}}
\def\tmu{\tilde{\mu}}
\def\tH{\tilde{H}}
\def\tdelta{\tilde{\delta}}
\def\tfrac#1#2{{\textstyle{#1\over#2}}}
\def\dfrac#1#2{{\displaystyle{#1\over#2}}}
%  \def\12{\frac 12}  removed by Henry -  conflict
\def\LR{{\rlap{\rm I}\hskip0.15em\hbox{\rm R}}}
\def\LZ{{\rlap{\rm Z}\hskip0.15em\hbox{\rm Z}}}
\def\LN{{\rlap{\rm I}\hskip0.2em\hbox{\rm N}}}
\def\f2{\frac{r}{2}}
\def\cO{{\cal O}}
\def\cP{{\cal P}}
\def\cQ{{\cal Q}}
\def\cA{{\cal A}}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\cF{{\cal F}}
\def\cL{{\cal L}}
\def\cB{{\cal B}}
\def\cN{{\cal N}}
\def\cS{{\cal S}}
\def\cT{{\cal T}}
%\def\cR{{\cal R}}
\def\Dx{\Delta x}
\def\Dy{\Delta y}
\def\Ds{\Delta s}
\def\D{\overline{\Delta}}
%\def\dx{\overline{\Delta x}}
%\def\dy{\overline{\Delta y}}
%\def\ds{\overline{\Delta s}}
%\def\svec{\mbox{\bf svec}}
\def\dx{p_x}
%\def\dy{p_y}
\def\ds{p_s}
\def\dv{p_v}
\def\bv{\bar v }
\def\bw{\bar w }
\def\bomega{\bar \omega }
\def\bomegas{\bar \omega^* }
\def\vs{v^*}
\def\bvs{\bv^*}
\def\iff{\mbox{ if and only if }}
\def\l{\left}
\def\r{\right}
\def\diag{{\rm diag}\,}
\def\supp{{\rm supp}\,}
\def\cond{{\rm cond}\,}
\def\oD{\overline{D}}
\def\oP{\overline{P}}
\def\omu{\overline{\mu}}
\def\norm#1{\l\|#1\r\|}
\def\infnorm#1{\l\|#1\r\|_\infty}
\def\br#1{\l(#1\r)}
\def\dd{\;:\;}
\def\k{,\;}
\def\a{\alpha}
\def\b{\beta}
\def\c{\gamma}
\def\g{\gamma}
\def\d{\delta}
\def\k{\kappa}
\def\s{\sigma}
\def\t{\vartheta}
\def\ll{\lambda}
\def\lll{<<}
\def\limsup{{\rm limsup}\,}
\def\hu{\hat u}
\def\e{\epsilon}
\def\bareps{\bar{\epsilon}}
\def\tv{\tilde{v}}

\def\mini#1{\mbox{\rm\small min}(#1)}
\def\maxi#1{\mbox{\rm\small max}(#1)}
\def\bvmin{\mini{\bv}}
\def\bvmax{\maxi{\bv}}
\def\bvsmin{\mini{\bar{v}^*}}
\def\bvsmax{\maxi{\bar{v}^*}}
\def\tvmin{\mini{\tilde{v}}}
\def\tvmax{\maxi{\tilde{v}}}
\def\vmin{\mini{v}}
\def\vmax{\maxi{v}}
\def\wmin{\mini{w}}
\def\wmax{\maxi{w}}
\def\vsmin{\mini{v^*}}
\def\set#1{\l\{#1\r\}}
\def\abs#1{\l| #1 \r|}

% From Nesterov - begins
\def\epi{{\rm epi \,}}
\def\cl{{\rm Cl \,}}
\def\con{{\rm Con \,}}
%\def\A{\mid }
\def\nr{\par \noindent}
\def\diag{{\rm diag \,}}
\def\im{{\rm Im \,}}
%\def\inter{{\rm int \,}}
\def\dom{{\rm dom \,}}
%\def\beq{\begin{equation}}
%\def\eeq{\end{equation}}
\newcommand{\rint}{{\rm rint\,}}
\newcommand{\NN}{\parallel}   % changed by Henry from \N

%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{corollary}{Corollary}[section]
%\newtheorem{conjecture}{Conjecture}[section]
%\newtheorem{proposition}{Proposition}[section]
%\newtheorem{assumption}{Assumption}[section]
%\newtheorem{definition}{Definition}[section]
%\newtheorem{property}{Property}[section]
%\newtheorem{example}{Example}[section]
%\newtheorem{remark}{Remark}[section]

%\newcommand{\proof}{\bf Proof: \rm \nr}
%\newcommand{\qed}{\hfill $\Box$ \nr \medskip}
\renewcommand\arraystretch{1}
%\renewcommand\arraystretch{1}
%\renewcommand\arraystretch{1}
%\def\ba{\begin{array}}
%\def\ea{\end{array}}
\def\beann{\begin{eqnarray*}}
\def\eeann{\end{eqnarray*}}
%\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}

\def\BT{\begin{theorem}}
\def\ET{\end{theorem}}
\def\BA{\begin{assumption}}
\def\EA{\end{assumption}}
\def\BD{\begin{definition}}
\def\ED{\end{definition}}
\def\BL{\begin{lemma}}
\def\EL{\end{lemma}}
\def\BC{\begin{corollary}}
\def\EC{\end{corollary}}
\def\BE{\begin{example}}
\def\EE{\end{example}}


%\def\la{\langle}
%\def\ra{\rangle}

\def\theequation{\thesection.\arabic{equation}}

% From Katya - begins
\newcommand{\Drawing}[3]{\mbox{\rule{0in}{#3}\special{psfile=#1}\hspace{#2}}}
\newcommand{\PSbox}[3]{\mbox{\rule{0in}{#3}\special{psfile=#1}\hspace{#2}}}
\newcommand{\PSframe}[3]{\framebox{\rule{0in}{#3}\special{psfile=#1}\hspace{#2}}
}
\newcommand{\ron}{{\bf R}_{_N}}
\newcommand{\rop}{{\bf R}_{_P}}
\newcommand{\rod}{{\bf R}_{_D}}
\newcommand{\rnk}{{\rm rank\,}}
\newcommand{\op}{{\bf O}_P}
\newcommand{\od}{{\bf O}_D}
\newcommand{\qop}{Q_{_P}}
\newcommand{\qod}{Q_{_D}}
\newcommand{\qn}{Q_{_N}}
\newcommand{\qond}{Q_{_{ND}}}
\newcommand{\qonp}{Q_{_{PN}}}
% From Serge Kruk
\newcommand{\minpgm}[3]{\mbox{#1~~~~~}\min\,\Bigl\{#2 \bigm| #3\Bigr\}}
\newcommand{\maxpgm}[3]{\mbox{#1~~~~~}\max\,\Bigl\{#2 \bigm| #3\Bigr\}}
\newcommand{\minpgms}[2]{\min\,\Bigl\{#1 \bigm| #2\Bigr\}}
\newcommand{\maxpgms}[2]{\max\,\Bigl\{#1 \bigm| #2\Bigr\}}
\newcommand{\maxmin}[3]{\max\,\Bigl\{\min\left\{#1\mid #2\right\}\bigm|#3\Bigr\}}
%\newcommand{\minmax}[3]{\min\,\Bigl\{\max\left\{#1\mid #2\right\}\bigm|#3\Bigr\}}

\newcommand{\order}[1]{{o}(#1)}

\def\Lag {{\cal L}}
\def\xk {{x}^{(k)}}
\def\hk {{d}^{(k)}}
\def\dk {{\delta}^{(k)}}
\def\xkp {{x}^{(k+1)}}
\def\hkp {{h}^{(k+1)}}
\def\xklk {({x}^{(k)},{\lambda}^{(k)})}
\def\delh{{\nabla h_j\xk}}
\def\delDh{{\nabla^2 h_j\xk}}
\def\delf{{\nabla f\xk}}
\def\delDf{{\nabla^2 f\xk}}
\def\delL {{\nabla {\cal L}\xklk}}
\def\delDL {{\nabla^2 {\cal L}\xklk}}
\def\zero {{\bf 0}}
\def\I    {{\bf I}}
\def\then {\Longrightarrow}
\def\eig {\mbox{\em Eig}}
\def\halmos{\hfill $\Box$}
%\newcommand{\diag}{{\rm diag\,}}
\newcommand{\Zero}[1]{{\bf 0}_{#1 \times #1}}
\newcommand{\Eye}[1]{{\bf I}_{#1 \times #1}}

\newcommand{\mybold}[1]{\textsl{#1}}
\def\sqp {\mybold{SQP}}
\def\sqqp {\mybold{SQ$^2$P}}
\def\nlp {\mybold{NLP}}
\def\unc {\mybold{UNC}}
\def\nep {\mybold{NEP}}
\def\foc {\mybold{FONS}}
\def\trs {\mybold{TRS}}
\def\nlind {NonLinDualSDP-TRS}
\def\lind {DSDP-TRS}
\def\linp {PSDP-TRS}
\def\trso {\mybold{TRS$_1$}}
\def\trst {\mybold{TRS$_2$}}
\def\eqtrs {\mybold{EQ-TRS}}
\def\cdt {\mybold{CDT}}
\def\ttrs {\mybold{2-TRS}}
\def\dttrs {\mybold{D2-TRS}}
\def\qqp {\mybold{Q$^2$P}}
\def\dqqp {\mybold{Dual-Q$^2$P}}
\def\nlpqp {\mybold{NLP-QP}}
\def\nlpqqp {\mybold{NLP-Q$^2$P}}
\def\sdp {\mybold{SDP}}
\def\qp {\mybold{QP}}
\def\psdp {\mybold{PSDP}}
\def\dsdp {\mybold{DSDP}}
\def\pqqp {\mybold{Projected\, Q$^2$P}}
\def\HQQP{H\qqp}
\def\HMQQP{HM\qqp}
\def\RE{\Re}
\def\Rn {\Re^n}
\def\Rm {\Re^m}
\def\SE {\cal S}
\def\SEn {\SE^{n+1}_+}
\def\fourth{\frac{1}{4}}
\def\threefourth{\frac{3}{4}}
\def\fivefourth{\frac{5}{4}}
\def\Lag {{\cal L}}
\newcommand{\Range}[1]{{\cal R}(#1)}
\newcommand{\Nullspace}[1]{{\cal N}(#1)}
\def\xk {x^{(k)}}
\def\lk {\lambda^{(k)}}
\def\lkp {\lambda^{(k+1)}}
\def\xl {x_\lambda}
\def\zl {z_\lambda}
\def\xs {\textbf{x}}
\def\df {:=}
\def\dft {~{\buildrel \triangle \over =}~}
\newcommand{\inner}[2]{{\langle #1 , #2 \rangle}}
\newcommand{\todo}[1]{{[\bf #1]}}
\newcommand{\Algot}[2]{
\begin{center}
  \fbox{\vbox{\baselineskip=7pt
      \begin{center}
        {\sc #1}
      \end{center}
      \begin{tabbing}
        #2
      \end{tabbing}}}
\end{center}}
%%% from Robustness chapter
\def\mypicts{\epsfxsize=120pt\epsfysize=120pt\epsffile}
\def\blcks{\hbox{\vrule width 4pt height 4pt depth 0pt}}
\def\blck{\hbox{\vrule width 6pt height 6pt depth 0pt}}
\def\geqq{\ge}
\def\blckl{\hbox{\vrule width 8pt height 8pt depth 0pt}}
\def\square{\hbox{\vrule\vbox{\hrule\phantom{o}\hrule}\vrule}}
%\def\blacksquare{\hbox{\vrule width 4pt height 4pt depth 0pt}}
\def\huh{\hbox{\vrule width 2pt height 8pt depth 2pt}}
\def\Bbb#1{{\bf #1}}
\def\lv{\left|\left|}
\def\rv{\right|\right|_2}
\def\Lin{{{\mathop{\rm Lin}}}}
\def\Span{{{\mathop{\rm Span}}}}
\def\cons{{{\mathop{\rm cons}}}}
\def\Row{{{\mathop{\rm Row}}}}
\def\V{{\mathop{{\rm V}}}}
\def\eqnum#1{\eqno (#1)}
% \def\fnote#1{\footnote}  % not used
\def\eint{$\bigcap$-ellipsoidal\ }
%\def\inter{\mathop{{\rm int}}}
\def\card{\mathop{{\rm card\,}}}
\def\nr{\par \noindent}
\def\ds{\displaystyle}
\def\ch{{\mathop{\rm cosh\,}}}
\def\sinh{{\mathop{\rm sinh}}}
\def\tgh{{\mathop{\rm tgh}}}
\def\chid{{\mathop{\chi}}}
\def\zet{{\mathop{\omega}}}
\def\Tr{\mathop{{\rm Tr}}}
\def\Sz{\mathop{{\hbox{\cal Size}}}}
%\def\Sz{\mathop{{\hbox{\sl Size}}}}   % canged to cal above
\def\nrm{\mathop{{\parallel}}}
\def\Det{\mathop{{\rm Det}\,}}
\def\dim{\mathop{{\rm dim}\,}}
%\def\inter{\mathop{{\rm int}\,}}
\def\vrt{\mathop{\,\mid}\,}
\def\pn{\par\noindent}
\def\argmin{\mathop{\rm argmin}}
%\def\cl{\mathop{{\rm cl}}}
\def\Dom{\mathop{{\rm Dom}}}
\def\Ker{\mathop{{\rm Ker}}}
\def\Argmin{\mathop{\rm Argmin}}
\def\Argmax{\mathop{\rm Argmax}}
\def\book{\cite{NNbook}$\,$}
\def\Kappa{{\mathop{\overline{\kappa}}}}
\def\Var{\mathop{{\rm V}}}
\def\tilde{\widetilde}
%%  \def\bar{\widehat}
\def\huh{\hbox{\vrule width 2pt height 8pt depth 2pt}}
\def\Bbb#1{{\bf #1}}
\def\Sep{{{\mathop{\rm Sep}}}}
\def\data{{{\mathop{\rm data}}}}
\def\Aff{{{\mathop{\rm Aff}}}}
%\def\n{{{\mathop{\rm n}}}}
\def\n{{{\mathop{0}}}}
\def\eqnum#1{\eqno (#1)}
%\def\fnote#1{\footnote}   % duplicated  and not used
%\def\blacksquare{\hbox{\vrule width 4pt height 4pt depth 0pt}}
\def\square{\hbox{\vrule\vbox{\hrule\phantom{o}\hrule}\vrule}}
%\def\inter{\mathop{{\rm int}}}
\def\SetUp{{{\mathop{{\cal B}_{\rm SetUp}}}}}
\def\SepK{{{\mathop{{\cal B}_{\rm SepK}}}}}
\def\RobF{{{\mathop{{\cal B}_{\rm InclK}}}}}
\def\setup{{{\mathop{{\cal A}_{\rm SetUp}}}}}
\def\SepU{{{\mathop{{\cal A}_{\rm Sep}}}}}
\def\epi{{\rm epi}}
\def\nr{\par \noindent}
\def\diag{{\rm diag}}
\def\Var{{{\mathop{\rm Var}}}}
\def\clsr{{\rm cl}}
%\def\beq{\begin{equation}}
%\def\eeq{\end{equation}}
\def\pn{\par\noindent}
%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{lemma}{Lemma}[section]
%\newtheorem{corollary}{Corollary}[section]
%\newtheorem{conjecture}{Conjecture}[section]
%\newtheorem{proposition}{Proposition}[section]
%\newtheorem{assumption}{Assumption}[section]
%\newtheorem{definition}{Definition}[section]
%\newtheorem{remark}{Remark}[section]
%\newcommand{\proof}{\bf Proof: \rm \nr}
%\newcommand{\qed}{\hfill $\Box$ \nr \medskip}
%\def\ba{\begin{array}}
%\def\ea{\end{array}}
%\def\beann{\begin{eqnarray*}}
%\def\eeann{\end{eqnarray*}}
%\def\bea{\begin{eqnarray}}
%\def\eea{\end{eqnarray}}
%\def\Tr{\mathop{{\rm Tr}}}
%\def\Det{\mathop{{\rm Det}}}
%\def\Sz{\mathop{{\hbox{\sl Size}}}}
%\def\Vl{\mathop{{\hbox{\sl Vol}}}}
%\def\CE{\mathop{{\hbox{\sl CircEll}}}}
%\def\CS{\mathop{{\hbox{\sl CircSimp}}}}
%\def\IE{\mathop{{\hbox{\sl InscrEll}}}}
\def\nrm{\mathop{{\parallel}}}
%\def\Det{\mathop{{\rm Det}\,}}
\def\dim{\mathop{{\rm dim}\,}}
%\def\inter{\mathop{{\rm int}\,}}
\def\vrt{\mathop{\,\mid}\,}
\def\argmin{\mathop{\rm argmin}}
%\def\cl{\mathop{\rm cl}\,}
\def\Dom{\mathop{\rm Dom}\,}
\def\4quad{\quad\quad\quad\quad}
\def\3quad{\quad\quad\quad}
\def\sign{{{\mathop{\rm sign}}}}
\def\Card{{{\mathop{\rm Card}}}}
\def\AA{{{\mathop{\cal A}}}}
\def\B{{{\mathop{\cal B}}}}
\def\C{{{\mathop{\cal C}}}}
\def\D{{{\mathop{\cal D}}}}

%%%%%%%%% my defs (LEG) %%%%%%%%%%%
\def\Dinf{{\mathcal D}_\infty}
\def\reals{\mbox{\bf R}}
\def\complex{\mbox{\bf C}}
\newcommand{\ie}{{\it i.e.}}
\newcommand{\eg}{{\it e.g.}}
\newcommand{\BEC}{\[\begin{array}{c}}
\newcommand{\EEC}{\end{array}\]}
\def\BEQ{\begin{equation}}
\def\EEQ{\end{equation}}
\def\sigmamin{{\sigma_{\rm min}}}
%**** Christoph Helmbergs definitions
\newcommand{\ip}[2]{\left\langle {#1},{#2}\right\rangle}
\def\Argmax{\mathop{\rm Argmax}}
\def\maxim{\mathop{\hbox{\rm maximize}}}
% *** Valerii Fedorov
%\newtheorem{valtheorem}{Theorem}
%\newcommand{\supp}{{\rm supp}}
%\newcommand{\rank}{{\rm rank}}
\newcommand{\EEE}{{\rm E}}
\newcommand{\EEEE}{{\cal E}}
%\newcommand{\Var}{{\rm Var}}
\newcommand{\Cov}{{\rm Cov}}
% *** from Gabor Pataki
%%3 new lines added july 21 - but commented out
\newcommand{\Omdown}{\mbox{$0  \,\,  0  \choose \,\, 0  \,\,  \Omega$}}
\newcommand{\ncone}{{\rm ncone \afterlog}}
\newcommand{\ggg}{\mbox{$ \,\,\,\, 4  \,\, -2 \,  \choose \,-2  \,\,\,\,\, 1 \,$}} % ( 4 -2; -2 1) 
%%end of 3 new lines added july 21
\newcommand{\mtxn}{{\cal R}^{n \times n}}
\newcommand{\symn}{{\cal S}^{n}}
\newcommand{\symr}{{\cal S}^{r}}
\newcommand{\radn}{{\cal R}^{n}}
\newcommand{\radm}{{\cal R}^{m}}
\newcommand{\radk}{{\cal R}^{k}}
\newcommand{\psdn}{{\cal S}^{n}_{+}}
\newcommand{\co}[1]{}
\newcommand{\bx}{\bar{x}}
\newcommand{\bX}{\bar{X}}
\newcommand{\by}{\bar{y}}
\newcommand{\bZ}{\bar{Z}}
\newcommand{\trileft}{\triangleleft}
\newcommand{\afterlog}{\,}
\newcommand{\lin}{{\rm lin \afterlog}}
\newcommand{\aff}{{\rm aff \afterlog}}
\newcommand{\face}{{\rm face \afterlog}}
\newcommand{\cone}{{\rm cone \afterlog}}
%\newcommand{\rank}{{\rm rank \afterlog}}
%\newcommand{\diag}{\rm diag \smallafterlog}
\newcommand{\mult}{\rm mult \smallsmallafterlog}
\newcommand{\rec}{\rm rec \afterlog}
\newcommand{\ri}{\rm ri \afterlog}
\newcommand{\rb}{\rm rb \afterlog}
\newcommand{\codim}{{\rm codim \afterlog}}
%\newcommand{\cl}{\rm cl \smallafterlog}
%\newcommand{\inter}{\rm int \afterlog}
\newcommand{\Ftri}{F^{\triangle} }
\newcommand{\Gtri}{G^{\triangle} }
\newcommand{\tri}{\triangle}
%\newcommand{\la}{\langle} 
%\newcommand{\ra}{\rangle} 
%\newcommand{\F}{{\cal F}}
%\newcommand{\A}{{\cal A}}
%\newcommand{\N}{{\cal N}}
\newcommand{\M}{{\cal M}}
\newcommand{\RR}{{\cal R}}
\newcommand{\HH}{{\cal H}}
\newcommand{\nin}{ \noindent }
%\newcommand{\R}{ {\cal R} }
%\newcommand{\beq}{\begin{equation}}
%\newcommand{\eeq}{\end{equation}}

\newcommand{\bex}{\begin{Example}}
\newcommand{\eex}{\end{Example}}

\newcommand{\bprop}{\begin{Proposition}}
\newcommand{\eprop}{\end{Proposition}}


\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}

\newcommand{\beqast}{\begin{eqnarray*}}
\newcommand{\eeqast}{\end{eqnarray*}}

\newcommand{\benum}{\begin{enumerate}}
\newcommand{\eenum}{\end{enumerate}}

\newcommand{\bitem}{\begin{itemize}}
\newcommand{\eit}{\end{itemize}}

\newcommand{\bth}{\begin{Theorem}}
\newcommand{\eth}{\end{Theorem}}

\newcommand{\ble}{\begin{Lemma}}
\newcommand{\ele}{\end{Lemma}}

\newcommand{\bcor}{\begin{Corollary}}
\newcommand{\ecor}{\end{Corollary}}

\newcommand{\brem}{\begin{Remark}}
\newcommand{\erem}{\end{Remark}}
%\newcommand{\bit}{\begin{itemize}}
%\newcommand{\eit}{\end{itemize}}
\newcommand{\dir}{\mbox{\rm dir }}
\newcommand{\tcone}{{\rm tcone \afterlog}}
\newcommand{\res}{\mbox{\rm res }}
\newcommand{\Proj}{\mbox{\rm Proj }}
\newcommand{\Lambdaup}{\mbox{$\Lambda  \,\,  0  \choose \,\, 0  \,\,  0$}}
\newcommand{\Iup}{\mbox{$I  \,\,  0  \choose \,\, 0  \,\,  0$}}
%\newcommand{\Ftri}{F^{\triangle} }
%\newcommand{\Gtri}{G^{\triangle} }
\newcommand{\Idown}{\mbox{$0  \,\,  0  \choose \,\, 0  \,\,  I$}}
\newcommand{\scrT}{\mbox{\tiny T}}




%% Added by Gabor on June 10. !!!!!
\newcommand{\bnd}{{\rm bnd \afterlog}}
\newcommand{\aaa}{\mbox{$\,\,\,\, 0  \,\,  -1 \, \choose \,  -1  \,\,\,\,\,  0 \,$}} % ( 0 -1 ; -1 0 )
%%%% \bbb changed to \bbh  by Henry
\newcommand{\bbh}{\mbox{$\, -1  \,\,\,\,\,  0  \, \choose \,\,\,\, 0  \,\,\,\,\,  0 \,$}} % ( -1 0 ; 0 0 )

\newcommand{\ccc}{\mbox{$ \,\,\,\, 0  \,\, -1 \,  \choose \, -1 \,\,\,\,\,  1  \,$}} % ( 0 -1; -1 1 )
\newcommand{\ddd}{\mbox{$ \,\,\,\, 0  \,\,\,\,\,  1 \,  \choose \,\,\,\, 1  \,\,\,\,\, 1 \,$}}   % ( 0 1 ; 1 1 )

\newcommand{\eee}{\mbox{$ \,\,\,\, 1  \,\,\,\,\,  1 \,  \choose \,\,\,\, 1  \,\,\,\,\, 1 \,$}} % ( 1 1; 1 1 )
\newcommand{\fff}{\mbox{$ \,\,\,\, 1  \,\,  -1 \,  \choose \, -1  \,\,\,\,\, 1 \,$}} % ( 1 -1 ; -1 1 )

\newcommand{\gggg}{\mbox{$ \,\,\,\, 4  \,\, -2 \,  \choose \,-2  \,\,\,\,\, 1 \,$}} % ( 4 -2; -2 1) 
\newcommand{\hhh}{\mbox{$ \,\,\,\, 4  \,\,\,\,\,  0 \,  \choose \,\,\,\, 0  \,\,\,\,\, 1 \,$}} % ( 4 0 ; 0 1 )

\newcommand{\iii}{\mbox{$ \,\,\,\, 1  \,\,\,\,\,  2 \,  \choose \,\,\,\,  2  \,\,\,\,\, 4 \,$}} % ( 1 2; 2 4 ) 

\newcommand{\jjj}{\mbox{$ \,\,\,\, 1  \,\,\,\,\,  0 \,  \choose \,\,\,\, 0  \,\,\,\,\, 1 \,$}} % ( 1 0 ; 0 1 )
\newcommand{\kkk}{\mbox{$ \,\,\,\, 1  \,\,\,\,\,  0 \,  \choose \,\,\,\, 0  \,\,-1 \,$}} % ( 1 0 ; 0 -1 )
\newcommand{\LLL}{\mbox{$ \,\,\,\, 0  \,\,\,\,\,  1 \,  \choose \,\,\,\, 1  \,\,\,\,\, 0 \,$}} % ( 0 1  ; 1 0  )

\newcommand{\evk}{\mbox{$EV_{k}$}}
\newcommand{\sdpk}{\mbox{$SDP_{\,k}$}}
\newcommand{\radd}{{\cal R}^{d}}

% *** from Gabor Pataki ends

%  from Florian's second contribution
\def\ca{{\cal A}}
\def\nz{\hfill\break}
\def\bbbh{b}
\def\mrr#1{$(\ref{#1})$}



%%  from Dimitris  oct 24 new start
\def\dimby{{\bf y}}
\def\dimbd{{\bf d}}
\newcommand{\mb}[1]{\mbox{\boldmath $#1$}}
\def\texitem#1{\par\smallskip\noindent\hangindent 25pt
               \hbox to 25pt {\hss #1 ~}\ignorespaces}
%%  from Dimitris  oct 24 new end
%%  from Dimitris  Qsymbols.tex file  - modified by Henry
% MATH SUMBOLS AND RELATED MACROS
%\def\a{\alpha}
%\def\b{\beta}

%\def\e{\epsilon}
%

%\def\xb{{\overline x}}
%\def\B{\overline{B}}
%\def\c{\overline{c}}
%\def\pibar{{\overline \pi}}
%\def\oS{{\overline S}}
%
%%\def\ba{{\bf a}}
\def\dimba{{\bf a}}
\def\bb{{\bf b}}    % needed
\def\dimbc{{\bf c}}   %  changed by Henry
\def\bd{{\bf d}}       %  needed
\def\dimbe{{\bf e}}     %  changed by Henry
\def\bff{{\bf f}}     % needed
%\def\bg{{\bf g}}
%\def\bh{{\bf h}}
%\def\bi{{\bf i}}
%\def\bj{{\bf j}}
\def\bk{{\bf k}}      %  needed
%\def\bl{{\bf l}}
%\def\bm{{\bf m}}
%\def\bbn{{\bf n}}
\def\bo{{\bf 0}}      %  needed
\def\bp{{\bf p}}   %  needed
\def\bq{{\bf q}}   % needed
\def\dimbr{{\bf r}}   % changed by Henry
%\def\bs{{\bf s}}
%\def\bt{{\bf t}}
%\def\bu{{\bf u}}
%\def\bv{{\bf v}}
%\def\bw{{\bf w}}
\def\dimbx{{\bf x}}    %  changed by Henry
%\def\by{{\bf y}}
\def\bz{{\bf z}}    %   needed
\def\bA{{\bf A}}    %  needed
%\def\bB{{\bf B}}
%\def\bC{{\bf C}}
%\def\bD{{\bf D}}
%\def\bE{{\bf E}}
%\def\bF{{\bf F}}
%\def\bG{{\bf G}}
\def\bH{{\bf H}}    % needed
\def\bI{{\bf I}}      %  needed
%\def\bK{{\bf K}}
%\def\bL{{\bf L}}
\def\bM{{\bf M}}     %  needed
%\def\bN{{\bf N}}
%\def\bO{{\bf O}}
\def\bP{{\bf P}}     %needed
\def\bQ{{\bf Q}}
\def\bR{{\bf R}}   %  needed
%\def\bS{{\bf S}}
%\def\bU{{\bf U}}
%\def\bT{{\bf T}}
\def\bY{{\bf Y}}        % needed
%\def\bX{{\bf X}}
%\def\bV{{\bf V}}
%\def\bW{{\bf W}}
%\def\bZ{{\bf Z}}
%\def\bo{{\bf 0}}
%%\def\bpi{{\bf p}}
\def\diml{{\mbox{\boldmath $\lambda$}}}
%\def\g{{\mbox{\boldmath $\gamma$}}}
%\def\G{{\mbox{\boldmath $\Gamma$}}}
\def\GG{{\mbox{\boldmath $\Gamma$}}}   % changed to GG from G by Henry
%\def\d{{\mbox{\boldmath $\delta$}}}
\def\dimL{{\mbox{\boldmath $\Lambda$}}}
%\def\a{{\mbox{\boldmath $\alpha$}}}
%\def\m{{\mbox{\boldmath $\mu$}}}
\def\dimr{{\mbox{\boldmath $\rho$}}}   %  changed by Henry
%\def\p{{\mbox{\boldmath $\pi$}}}
%\def\th{{\mbox{\boldmath $\theta$}}}
%\def\x{{\mbox{\boldmath $\xi$}}}
%\def\b{{\mbox{\boldmath $\beta$}}}
%\def\z{{\mbox{\boldmath $\zeta$}}}
%\def\s{{\mbox{\boldmath $\sigma$}}}
%\def\S{{\mbox{\boldmath $\Sigma$}}}
\def\SS{{\mbox{\boldmath $\Sigma$}}}  % changed by Henry to SS from S
%
%\def\boc{{\bf \overline c}}
%\def\bxb{{\bf \overline x}}
%\def\bxbar{{\bf \overline x}}
%\def\bybar{{\bf \overline y}}
%\def\bxhat{{\bf \hat x}}
%\def\bxx{{\bf \overline x}}
\def\bbb{{\bf \overline b}}
%\def\cb{{\bf \overline c}}
%\def\Bb{{\bf \overline B}}
%\def\bDelta{{\bf D}}
%\def\bfbar{{\bf \overline f}}
%\def\bfhat{{\bf \hat f}}
%\def\bftilde{{\bf \tilde f}}
%\def\xbar{{\overline x}}
%\def\vbar{{\overline v}}
%\def\Gbar{{\overline G}}
%\def\Ebar{{\overline E}}
%\def\xhat{{\hat x}}
%\def\At{{\bf \tilde A}}
%\def\bt{{\bf \tilde b}}
%\def\fbar{{\overline f}}
%\def\fhat{{\hat f}}
%\def\Lbar{{\overline L}}
%
%\def\Ascr{{\cal A}}     \def\A{{\cal A}}
%\def\Bscr{{\cal B}}
%\def\Cscr{{\cal C}}
%\def\Fscr{{\cal F}}
%\def\N{{\cal N}}
%\def\E{{\cal E}}
%\def\Pscr{{\cal P}}
%\def\Xscr{{\cal X}}
%\def\Yscr{{\cal Y}}
%\def\Zscr{{\cal Z}}
%
%  From Ragu  May 3
\def\Co{\mathop{\mbox{\bf Co}}}
\def\complexragu{{\bf C}}   % changed by Henry
\def\reals{{\mbox{\bf R}}}
\def\bD{{\mathbf{\Delta}}} 
\def\bDelta{{\mathbf{\Delta}}} 
\def\bPifin{{\bPi}_{\rm fin}}
\def\bTheta{{\bf \Theta}}
\def\Ltwo{{\bf L}_2} 
\def\bPi{\mathbf{\Pi}}
%\def\rank{\mbox{rank}}

\newcommand{\AR}[2]{\left[\begin{array}{#1}#2\end{array}\right]}
%%  end from Ragu




%%%%%  newcommands from Alizadeh 
%\newcommand{\zup}{\overline{z}}
%\newcommand{\zdown}{\underline{z}}
%\newcommand{\phiup}{\overline{\phi}}
%\newcommand{\phidown}{\underline{\phi}}
%\newcommand{\Xhalf}{X^{1/2}}
%\newcommand{\Xnhalf}{X^{-1/2}}
%\newcommand{\VEC}{{\bf vec}}
%\newcommand{\MAT}{{\rm Mat}}

%\newcommand{\rk}{\operatornamewithlimits{rk}}   %changed by Henry below
%\newcommand{\rk}{\rm rank}
%\newcommand{\jrk}[1]{\rk_{#1}}
%\newcommand{\rank}{\operatorname{rank}}
%\newcommand{\Arw}{\operatorname{Arw}}   %  changed to below by Henry
%\newcommand{\Arw}{{\cal Arw}}
%\newcommand{\Ocomm}{C_{\J}}
%\newcommand{\Comm}{\mathfrak{C}}   %  commented out by Henry
%\newcommand{\bigO}{\mathcal{O}}
%\newcommand{\Cn}{\mathcal{C}_n}
%\newcommand{\defeq}{\stackrel{\mathrm{def}}{=}}
%\newcommand{\pd}{\succ}
%\nc{\VV}{\mbox{\bf vec\hspace{.03cm}}}
%\newcommand{\xbar}{\bar{x}}
%\newcommand{\ybar}{\bar{y}}
%\newcommand{\zbar}{\bar{z}}
%\newcommand{\bvv}[1]{\bar{\mathbf{#1}}}
%\newcommand{\Th}[1]{$#1^\mathrm{th}$}
%\newcommand{\ubar}{\bar{u}}
%\newcommand{\tlup}[1]{\widetilde{#1}}
%\newcommand{\tldn}[1]{\undertilde{#1}}
%\newcommand{\Int}{\operatorname{Int}}    % commented out by Henry to ??
%%%%%%%%%%%%%%%%%%%%????????
%\newcommand{\sgn}{\operatornamewithlimits{sgn}}
%\newcommand{\vzero}{\mathbf{0}}
%\newcommand{\vone}{\mathbf{1}}
%\newcommand{\va}{\mathbf{a}}
%\newcommand{\vb}{\mathbf{b}}
%\newcommand{\vc}{\mathbf{c}}
%\newcommand{\vd}{\mathbf{d}}
%\newcommand{\ve}{\mathbf{e}}
%\newcommand{\vf}{\mathbf{f}}
%\newcommand{\vg}{\mathbf{g}}
%\newcommand{\vp}{\mathbf{p}}
%\newcommand{\vq}{\mathbf{q}}
%\newcommand{\vt}{\mathbf{t}}
%\newcommand{\vu}{\mathbf{u}}
%\newcommand{\vv}{\mathbf{v}}
%\renewcommand{\vv}{\mathbf{v}}
%\newcommand{\vvA}{\mathbf{v}}
%\newcommand{\vw}{\mathbf{w}}
%\newcommand{\vx}{\mathbf{x}}
%\newcommand{\vy}{\mathbf{y}}
%\newcommand{\vz}{\mathbf{z}}
%\newcommand{\xt}{\tlup{\vx}}
%\newcommand{\wt}{\tlup{\vw}}
%\newcommand{\At}{\tldn{A}}
%\newcommand{\at}{\tldn{\va_i}}
%\newcommand{\Xt}{\tilde{X}}
%\newcommand{\ct}{\tldn{\vc}}
%\newcommand{\etA}{\tldn{\ve}}
%\newcommand{\st}{\tldn{\vsA}}
%\newcommand{\St}{\tldn{S}}
%\newcommand{\dXt}{\widetilde{\dX}}
%\newcommand{\dSt}{\tldn{\dS}}
%\newcommand{\dxt}{\tlup{\dx}}
%\newcommand{\dst}{\tldn{\ds}}
%\newcommand{\onehalf}{{1\!/\!2}}
%\newcommand{\nonehalf}{{-\!1\!/\!2}}
%\newcommand{\xtsq}{\xt^{\onehalf}}
%\newcommand{\xtisq}{\xt^{\nonehalf}}
%\newcommand{\stsq}{\st^{\onehalf}}
%\newcommand{\stisq}{\st^{\nonehalf}}
%\newcommand{\Xtsq}{\Xt^{\onehalf}}
%\newcommand{\Xtisq}{\Xt^{\nonehalf}}
%\newcommand{\Stsq}{\St^{\onehalf}}
%\newcommand{\Stisq}{\tldn{S}^{\nonehalf}}
%\newcommand{\Xsq}{X^{\onehalf}}
%\newcommand{\Xisq}{X^{\nonehalf}}
%\newcommand{\Ssq}{S^{\onehalf}}
%\newcommand{\Sisq}{S^{\nonehalf}}
%\newcommand{\xsq}{\vx^{\onehalf}}
%\newcommand{\xisq}{\vx^{\nonehalf}}
%\newcommand{\ssq}{\vsA^{\onehalf}}
%\newcommand{\sisq}{\vsA^{\nonehalf}}
%\newcommand{\ah}{\underaccent{\hat}{\va_i}}
%\newcommand{\Ah}{\underaccent{\hat}{A}}
%\newcommand{\sh}{\hat{\vsA}}
%\newcommand{\xh}{\hat{\vx}}
%\newcommand{\yh}{\hat{\vy}}
%\newcommand{\hx}{\hat{x}_j}
%\newcommand{\norminf}[1]{\left\|{#1}\right\|_{\infty}}
%\newcommand{\normtwo}[1]{\left\|{#1}\right\|_2}
%\newcommand{\normF}[1]{\left\|{#1}\right\|_{\mathrm{F}}}
%\newcommand{\normP}[1]{\left\|{#1}\right\|_{\mathrm{p}}}
%\newcommand{\jnormF}[2]{\left\| {#1}\right\|_{\mathrm{F/{#2}}}}
%\newcommand{\scp}[2]{\langle{#1},{#2}\rangle}
%\newcommand{\jscp}[3]{\langle{#1},{#2}\rangle_{#3}}
%\newcommand{\jdet}[1]{\det_{#1}}
%\newcommand{\jtr}[1]{\tr_{#1}}
%\newcommand{\Bn}{\mathcal{B}_n}
%\newcommand{\Gp}{\mathcal{G}_+}
%\newcommand{\Qp}[1]{\mathbf{Q}_{#1}}
%\newcommand{\jprod}{\circ}
%\newcommand{\J}{\mathcal{J}}
%\newcommand{\J}{\mathfrak{J}}    %  commented out by Henry
%\newcommand{\Lo}{\mathcal{L}}
%\newcommand{\ProjA}{\mathcal{P}}
%\newcommand{\Q}{\mathcal{Q}}
%\newcommand{\Oc}{\mathcal{O}}
%\newcommand{\Sy}{\mathcal{S}}
%\newcommand{\bbC}{\mathbb{C}}
%\newcommand{\bbH}{\mathbb{H}}
%\newcommand{\bbM}{\mathbb{M}}
%\newcommand{\bbO}{\mathbb{O}}
%\newcommand{\bbQ}{\mathbb{Q}}
%\newcommand{\mathA}{\mathfrak{A}}    %  changed to following by Henry
%\newcommand{\mathA}{\A}
%\newcommand{\mathC}{\A}
%\newcommand{\mathH}{\A}
%\newcommand{\mathM}{\A}
%\newcommand{\mathO}{\A}
%\newcommand{\mathQ}{\A}
%\newcommand{\mathS}{\A}
%\newcommand{\mathC}{\mathfrak{C}}   %  changed till math S below
%\newcommand{\mathH}{\mathfrak{H}}
%\newcommand{\mathM}{\mathfrak{M}}
%\newcommand{\mathO}{\mathfrak{O}}
%\newcommand{\mathQ}{\mathfrak{Q}}
%\newcommand{\mathS}{\mathfrak{S}}
%\newcommand{\ST}{{\mbox{s.t.}}}
%\newcommand{\NF}[1]{\N_{F/{#1}}}
%\newcommand{\Quart}{\mbox{\boldmath H}}
%\newcommand{\psd}{\succcurlyeq}   %  changed by Henry
%\newcommand{\psd}{\succeq}
%\newcommand{\pdef}{\succ}
%\newcommand{\tr}{\operatorname{tr}}
%\renewcommand{\tr}{\operatorname{tr}}   % commented out by Henry
%\newcommand{\Det}{\operatorname{Det}}
%\renewcommand{\Det}{\operatorname{Det}}
%\newcommand{\Tr}{\operatorname{Tr}}
%\renewcommand{\Tr}{\operatorname{Tr}}  %  changed by Henry to above
%             i.e. commented out
%\newcommand{\A}{\mathcal{A}}
%\renewcommand{\A}{\mathcal{A}}
%\newcommand{\B}{\mathcal{B}}
%\renewcommand{\B}{\mathcal{B}}
%\newcommand{\C}{\mathcal{C}}
%\renewcommand{\C}{\mathcal{C}}
%\newcommand{\E}{\mathcal{E}}
%\renewcommand{\E}{\mathcal{E}}
%\newcommand{\F}{\mathcal{F}}
%\renewcommand{\F}{\mathcal{F}}
%\newcommand{\G}{\mathcal{G}}
%\renewcommand{\G}{\mathcal{G}}
%\renewcommand{\H}{\mathcal{H}}
%\newcommand{\N}{\mathcal{N}}
%\renewcommand{\N}{\mathcal{N}}
%\newcommand{\K}{\mathcal{K}}
%\renewcommand{\K}{\mathcal{K}}
%\newcommand{\M}{\mathcal{M}}
%\newcommand{\J}{\mathcal{J}}
%\renewcommand{\M}{\mathcal{M}}
%\newcommand{\T}{\mathcal{T}}
%\renewcommand{\T}{\mathcal{T}}
%\newcommand{\vs}{\mathbf{s}}
%\newcommand{\vsA}{\mathbf{s}}
%\newcommand{\dx}{\Delta \vx}
%\newcommand{\dxA}{\Delta \vx}
%\newcommand{\dy}{\Delta \vy}
%\renewcommand{\dy}{\Delta \vy}
%\newcommand{\dyA}{\Delta \vy}
%\newcommand{\ds}{\Delta \vs}
%\renewcommand{\ds}{\Delta \vs}
%\newcommand{\dsA}{\Delta \vs}
%\newcommand{\ch}{\hat{\vc}}
%\renewcommand{\ch}{\underaccent{\hat}{\vc}}
%\newcommand{\chA}{\underaccent{\hat}{\vc}}


\newcommand{\zup}{\overline{z}}
\newcommand{\zdown}{\underline{z}}
\newcommand{\phiup}{\overline{\phi}}
\newcommand{\phidown}{\underline{\phi}}
\newcommand{\Xhalf}{X^{1/2}}
\newcommand{\Xnhalf}{X^{-1/2}}

%\newcommand{\jrk}{\operatornamewithlimits{rk}}
\newcommand{\jrk}{\mathrm{rk\:}}
%\newcommand{\cork}{\operatornamewithlimits{cork}}
\newcommand{\cork}{\mathrm{cork\:}}
%
%\newcommand{\jrk}[1]{\rk_{#1}}
\newcommand{\Arw}{\mathbf{Arw\:}}
%\newcommand{\Ocomm}{C_{\mathJ}}
\newcommand{\Comm}{\mathfrak{C}}
\newcommand{\bigO}{\mathcal{O}}
\newcommand{\Cn}{\mathcal{C}_n}
\newcommand{\defeq}{\stackrel{\mathrm{def}}{=}}
\newcommand{\pd}{\succ}
\newcommand{\VV}{\mbox{\bf vec\hspace{.03cm}}}
\newcommand{\xbar}{\bar{x}}
\newcommand{\ybar}{\bar{y}}
\newcommand{\zbar}{\bar{z}}
\newcommand{\bvv}[1]{\bar{\mathbf{#1}}}
\newcommand{\Th}[1]{$#1^\mathrm{th}$}
\newcommand{\ubar}{\bar{u}}
\newcommand{\tlup}[1]{\widetilde{#1}}
\newcommand{\tldn}[1]{\undertilde{#1}}
%\newcommand{\Int}{\operatorname{Int}}
\newcommand{\Interior}{\mathrm{Int\:}}
\newcommand{\veczero}{\mathbf{0}}
\newcommand{\vecone}{\mathbf{1}}
\newcommand{\veca}{\mathbf{a}}
\newcommand{\vecb}{\mathbf{b}}
\newcommand{\vecc}{\mathbf{c}}
\newcommand{\vecd}{\mathbf{d}}
\newcommand{\vece}{\mathbf{e}}
\newcommand{\vecf}{\mathbf{f}}
\newcommand{\vecg}{\mathbf{g}}
\newcommand{\vecp}{\mathbf{p}}
\newcommand{\vecq}{\mathbf{q}}
\newcommand{\vecs}{\mathbf{s}}
\newcommand{\vect}{\mathbf{t}}
\newcommand{\vecu}{\mathbf{u}}
\newcommand{\vecv}{\mathbf{v}}
\newcommand{\vecw}{\mathbf{w}}
\newcommand{\vecx}{\mathbf{x}}
\newcommand{\vecy}{\mathbf{y}}
\newcommand{\vecz}{\mathbf{z}}
\newcommand{\xt}{\tlup{\vecx}}
\newcommand{\wt}{\tlup{\vecw}}
\newcommand{\At}{\tldn{A}}
\newcommand{\at}{\tldn{\veca_i}}
\newcommand{\Xt}{\tilde{X}}
\newcommand{\ct}{\tldn{\vecc}}
%\renewcommand{\et}{\tldn{\vece}}
\newcommand{\etilde}{\tldn{\vece}}
\newcommand{\st}{\tldn{\vecs}}
\newcommand{\St}{\tldn{S}}
\newcommand{\onehalf}{{1\!/\!2}}
\newcommand{\nonehalf}{{-\!1\!/\!2}}
\newcommand{\xtsq}{\xt^{\onehalf}}
\newcommand{\xtisq}{\xt^{\nonehalf}}
\newcommand{\stsq}{\st^{\onehalf}}
\newcommand{\stisq}{\st^{\nonehalf}}
\newcommand{\Xtsq}{\Xt^{\onehalf}}
\newcommand{\Xtisq}{\Xt^{\nonehalf}}
\newcommand{\Stsq}{\St^{\onehalf}}
\newcommand{\Stisq}{\tldn{S}^{\nonehalf}}
\newcommand{\Xsq}{X^{\onehalf}}
\newcommand{\Xisq}{X^{\nonehalf}}
\newcommand{\Ssq}{S^{\onehalf}}
\newcommand{\Sisq}{S^{\nonehalf}}
\newcommand{\xsq}{\vecx^{\onehalf}}
\newcommand{\xisq}{\vecx^{\nonehalf}}
\newcommand{\ssq}{\vecs^{\onehalf}}
\newcommand{\sisq}{\vecs^{\nonehalf}}
\newcommand{\ah}{\underaccent{\hat}{\veca_i}}
\newcommand{\Ah}{\underaccent{\hat}{A}}
\newcommand{\chat}{\hat{\vecc}}
\newcommand{\shat}{\hat{\vecs}}
\newcommand{\xhat}{\hat{\vecx}}
\newcommand{\yhat}{\hat{\vecy}}
\newcommand{\hx}{\hat{x}_j}
\newcommand{\norminf}[1]{\left\|{#1}\right\|_{\infty}}
\newcommand{\normtwo}[1]{\left\|{#1}\right\|_2}
\newcommand{\normF}[1]{\left\|{#1}\right\|_{\mathrm{F}}}
\newcommand{\normP}[1]{\left\|{#1}\right\|_{\mathrm{p}}}
\newcommand{\jnormF}[2]{\left\| {#1}\right\|_{\mathrm{F/{#2}}}}
\newcommand{\scp}[2]{\langle{#1},{#2}\rangle}
\newcommand{\jscp}[3]{\langle{#1},{#2}\rangle_{#3}}
%\newcommand{\jtr}{\operatorname{tr}}
\newcommand{\jtr}{\mathrm{tr\:}}
%\newcommand{\jdet}[1]{\det_{#1}}
\newcommand{\jdet}{\det}
\newcommand{\Qp}[1]{\mathbf{Q}_{#1}}
\newcommand{\jprod}{\circ}
%\newcommand{\Bn}{\mathcal{B}_n}
%\newcommand{\Gp}{\mathcal{G}_+}
%\renewcommand{\H}{\mathcal{H}}
\newcommand{\Lo}{\mathcal{L}}
\newcommand{\Po}{\mathcal{P}}
\newcommand{\Q}{\mathcal{Q}}
\newcommand{\Oc}{\mathcal{O}}
\newcommand{\Sy}{\mathcal{S}}
\newcommand{\bbC}{\mathbb{C}}
\newcommand{\bbH}{\mathbb{H}}
\newcommand{\bbM}{\mathbb{M}}
\newcommand{\bbO}{\mathbb{O}}
\newcommand{\bbQ}{\mathbb{Q}}
\newcommand{\mathA}{\mathfrak{A}}
\newcommand{\mathC}{\mathfrak{C}}
\newcommand{\mathH}{\mathfrak{H}}
\newcommand{\mathJ}{\mathfrak{J}}
\newcommand{\mathM}{\mathfrak{M}}
\newcommand{\mathO}{\mathfrak{O}}
\newcommand{\mathQ}{\mathfrak{Q}}
\newcommand{\mathS}{\mathfrak{S}}
\newcommand{\ST}{{\mbox{s.t.}}}
\newcommand{\NF}[1]{\N_{F/{#1}}}
\newcommand{\Quart}{\mbox{\boldmath H}}
%\newcommand{\psd}{\succcurlyeq}
%\newcommand{\psd}{\succeq}    %%  changed by Henry again!!!
%\newcommand{\psd}{\underline{\succ}}    %%  changed by Henry again!!!
\newsymbol\psd 133C
\newcommand{\pdef}{\succ}
\newenvironment{FaridExample}[1]{\begin{example}[#1]\rm}{{\hfill\rule{2mm}{2mm}\par}\end{example}}
\newenvironment{FaridRemark}{\begin{remark}\rm}{{\hfill\rule{2mm}{2mm}\par}\end{remark}}
%\renewenvironment{proof}{{\noindent\textbf{Proof:}}}{\hfill\rule{2mm}{2mm}\par}
%

%%%%%  end from Alizadeh
%%%%%  from Sturm/Luo
\newenvironment{proofenv}{\bpr}{\QED}
\newenvironment{proofA}{\bpr}{\QED}
\newcommand{\epc}{\hspace{1pc}}
\newcommand{\SOC}{{\rm SOC}}
\newcommand{\PSD}{{\rm PSD}}
\newcommand{\OAut}{{\rm OAut}}
\def\cK{{\cal K}}
\newcommand{\Vec}{{\rm \;vec }}
\newcommand{\Mat}{{\rm \;mat }}
\newcommand{\Img}{{\rm \;Img\, }}
\newcommand{\Face}{{\rm \;face\, }}
\newcommand{\Rel}{{\rm \;relint\, }}
\newcommand{\Int}{{\rm \;int\, }}
\newcommand{\implies}{\Longrightarrow}
\newcommand{\Dim}{{\rm \;dim\, }}
\newcommand{\ifif}{\Longleftrightarrow}
\newcommand{\TransP}{T}   %% added to avoid ^\T conflict for transpose
%%%%%  endfrom Sturm/Luo
