\newcounter{optproblem}
\def \Tagtheproblem   {\refstepcounter{optproblem}\tag{P\arabic{optproblem}}}

%\theoremstyle{plain}
\newtheoremstyle{mytheoremstyle} % name
    {\topsep}                    % Space above
    {\topsep}                    % Space below
    {\normalfont}                % Body font
    {}                           % Indent amount
    %{\scshape}                   % Theorem head font
    {\bfseries}                   % Theorem head font
    {.}                          % Punctuation after theorem head
    {.5em}                       % Space after theorem head
    {}  % Theorem head spec (can be left empty, meaning â€˜normalâ€?)

\theoremstyle{mytheoremstyle}
%\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem*{theorem*}{Theorem}
\newtheorem*{lemma*}{Lemma}
\newtheorem*{remark*}{Remark}
\newtheorem{lemma}[theorem]{Lemma}%[section]
\newtheorem{property}[theorem]{Property}
\newtheorem{assumption}{Assumption}

%letter numbered
\newtheorem{ClaimX}{Claim}
\renewcommand{\theClaimX}{\Alph{ClaimX}} % "letter-numbered" lemmas
\newtheorem{LemmaX}{Lemma}
\renewcommand{\theLemmaX}{\Alph{LemmaX}} % "letter-numbered" lemmas
\newtheorem{CorollaryX}{Corollary}
\renewcommand{\theCorollaryX}{\Alph{CorollaryX}} % "letter-numbered" lemmas
\newtheorem{TheoremX}{Theorem}
\renewcommand{\theTheoremX}{\Alph{TheoremX}} % "letter-numbered" lemmas
\newtheorem{DefinitionX}{Definition}
\renewcommand{\theDefinitionX}{\Alph{DefinitionX}} % "letter-numbered" lemmas

\theoremstyle{mytheoremstyle}
\newtheorem{definition}{Definition}[section]

% \newenvironment{proofoutline}{\noindent{\bfseries Proof outline}\hspace*{1em}}{\qed\bigskip}

\newenvironment{proofoutline}
{\begin{proof}[Proof outline]}
{\end{proof}}



\theoremstyle{remark}
%\renewcommand{\qedsymbol}{\FourClowerSolid}

%===========================================================
\DeclareMathAlphabet{\pazocal}{OMS}{zplm}{m}{n}
\DeclareMathAlphabet{\mathpzc}{OMS}{pzc}{m}{it}

\setlist[itemize]{leftmargin=*}

%\setlength{ skip}{1pt}
%\setlength{ indent}{0pt}

%\def\[#1\]{\vspace{1mm}\begin{align}#1 \vspace{1mm}\end{align}}

% ========================
% Letter
% ========================
\renewcommand{\hat}{\widehat}
\def\blue#1{{\color{blue}#1}}
\def\red#1{{\color{red}#1}}


\newcommand{\bfm}[1]{\ensuremath{\mathbf{#1}}}
\newcommand{\bfsym}[1]{\ensuremath{\boldsymbol{#1}}}

\def\ba{\bfm a}   \def\bA{\bfm A}  \def\AA{\mathbb{A}}
\def\bb{\bfm b}   \def\bB{\bfm B}  \def\BB{\mathbb{B}}
\def\bc{\bfm c}   \def\bC{\bfm C}  \def\CC{\mathbb{C}}
\def\bd{\bfm d}   \def\bD{\bfm D}  \def\DD{\mathbb{D}}
\def\be{\bfm e}   \def\bE{\bfm E}  \def\EE{\mathbb{E}}
\def\bff{\bfm f}  \def\bF{\bfm F}  \def\FF{\mathbb{F}}
\def\bg{\bfm g}   \def\bG{\bfm G}  \def\GG{\mathbb{G}}
\def\bh{\bfm h}   \def\bH{\bfm H}  \def\HH{\mathbb{H}}
\def\bi{\bfm i}   \def\bI{\bfm I}  \def\II{\mathbb{I}}
\def\bj{\bfm j}   \def\bJ{\bfm J}  \def\JJ{\mathbb{J}}
\def\bk{\bfm k}   \def\bK{\bfm K}  \def\KK{\mathbb{K}}
\def\bl{\bfm l}   \def\bL{\bfm L}  \def\LL{\mathbb{L}}
\def\bm{\bfm m}   \def\bM{\bfm M}  \def\MM{\mathbb{M}}
\def\bn{\bfm n}   \def\bN{\bfm N}  \def\NN{\mathbb{N}}
\def\bo{\bfm o}   \def\bO{\bfm O}  \def\OO{\mathbb{O}}
\def\bp{\bfm p}   \def\bP{\bfm P}  \def\PP{\mathbb{P}}
\def\bq{\bfm q}   \def\bQ{\bfm Q}  \def\QQ{\mathbb{Q}}
\def\br{\bfm r}   \def\bR{\bfm R}  \def\RR{\mathbb{R}}
\def\bs{\bfm s}   \def\bS{\bfm S}  \def\SS{\mathbb{S}}
\def\bt{\bfm t}   \def\bT{\bfm T}  \def\TT{\mathbb{T}}
\def\bu{\bfm u}   \def\bU{\bfm U}  \def\UU{\mathbb{U}}
\def\bv{\bfm v}   \def\bV{\bfm V}  \def\VV{\mathbb{V}}
\def\bw{\bfm w}   \def\bW{\bfm W}  \def\WW{\mathbb{W}}
\def\bx{\bfm x}   \def\bX{\bfm X}  \def\XX{\mathbb{X}}
\def\by{\bfm y}   \def\bY{\bfm Y}  \def\YY{\mathbb{Y}}
\def\bz{\bfm z}   \def\bZ{\bfm Z}  \def\ZZ{\mathbb{Z}}
\def\bzero{\bfm 0} \def\bone{\bfm 1}

\def\calA{{\cal  A}} \def\cA{{\cal  A}}
\def\calB{{\cal  B}} \def\cB{{\cal  B}}
\def\calC{{\cal  C}} \def\cC{{\cal  C}}
\def\calD{{\cal  D}} \def\cD{{\cal  D}}
\def\calE{{\cal  E}} \def\cE{{\cal  E}}
\def\calF{{\cal  F}} \def\cF{{\cal  F}}
\def\calG{{\cal  G}} \def\cG{{\cal  G}}
\def\calH{{\cal  H}} \def\cH{{\cal  H}}
\def\calI{{\cal  I}} \def\cI{{\cal  I}}
\def\calJ{{\cal  J}} \def\cJ{{\cal  J}}
\def\calK{{\cal  K}} \def\cK{{\cal  K}}
\def\calL{{\cal  L}} \def\cL{{\cal  L}}
\def\calM{{\cal  M}} \def\cM{{\cal  M}}
\def\calN{{\cal  N}} \def\cN{{\cal  N}}
\def\calO{{\cal  O}} \def\cO{{\cal  O}}
\def\calP{{\cal  P}} \def\cP{{\cal  P}}
\def\calQ{{\cal  Q}} \def\cQ{{\cal  Q}}
\def\calR{{\cal  R}} \def\cR{{\cal  R}}
\def\calS{{\cal  S}} \def\cS{{\cal  S}}
\def\calT{{\cal  T}} \def\cT{{\cal  T}}
\def\calU{{\cal  U}} \def\cU{{\cal  U}}
\def\calV{{\cal  V}} \def\cV{{\cal  V}}
\def\calW{{\cal  W}} \def\cW{{\cal  W}}
\def\calX{{\cal  X}} \def\cX{{\cal  X}}
\def\calY{{\cal  Y}} \def\cY{{\cal  Y}}
\def\calZ{{\cal  Z}} \def\cZ{{\cal  Z}}
\def\bZero{\bfm 0}

% =========================
% Greek Letters
% =========================
\def\balpha{\bfsym \alpha}
\def\bbeta{\bfsym \beta}
\def\bgamma{\bfsym \gamma}
\def\bGamma{\bfsym \Gamma}
\def\bdelta{\bfsym {\delta}}
\def\bDelta {\bfsym {\Delta}}
\def\bfeta{\bfsym {\eta}}
\def\bfEta {\bfsym {\Eta}}
\def\bmu{\bfsym {\mu}}
\def\bMu {\bfsym {\Mu}}
\def\bnu{\bfsym {\nu}}
\def\bpi{\bfsym {\pi}}
\def\bPi{\bfsym {\Pi}}
\def\btheta{\bfsym {\theta}}
\def\bTheta {\bfsym {\Theta}}
\def\beps{\bfsym \varepsilon}
\def\bepsilon{\bfsym \varepsilon}
\def\bsigma{\bfsym \sigma}
\def\bSigma{\bfsym \Sigma}
\def\blambda {\bfsym {\lambda}}
\def\bLambda {\bfsym {\Lambda}}
\def\bomega {\bfsym {\omega}}
\def\bOmega {\bfsym {\Omega}}
\def\brho   {\bfsym {\rho}}
\def\btau{\bfsym {\tau}}

\def\bxi{\bfsym {\xi}}
\def\bXi{\bfsym {\Xi}}


\def\bzeta{\bfsym {\zeta}}
\def\bZeta{\bfsym {\zeta}}

\def\halpha{\hat{\alpha}}
\def\hbalpha{\hat{\bfsym \alpha}}

\def\hbeta{\hat{\beta}}
\def\hbbeta{\hat{\bfsym \beta}}

\def\hgamma{\hat{\gamma}}
\def\hgamma{\hat{\bfsym \gamma}}

\def\hGamma{\hat{ \Gamma}}
\def\hbGamma{\hat{\bfsym \Gamma}}

\def\hdelta{\hat{\delta}}
\def\hbdelta{\hat{\bfsym {\delta}}}

\def\hDelta {\hat{\Delta}}
\def\hbDelta{\hat{\bfsym {\Delta}}}

\def\heta{\hat {\eta}}
\def\hbfeta {\hat{\bfsym {\eta}}}

\def\hmu{\hat{\mu}}
\def\hbmu {\hat{\bfsym {\mu}}}

\def\hnu{\hat{\nu}}
\def\hbnu {\hat{\bfsym {\nu}}}

\def\htheta{\hat {\theta}}
\def\hbtheta {\hat{\bfsym {\theta}}}

\def\hTheta{\hat {\Theta}}
\def\hbTheta {\hat{\bfsym {\Theta}}}

\def\hbeps{\hat{\bfsym \varepsilon}}
\def\hbepsilon{\hat{\bfsym \varepsilon}}

\def\hsigma{\hat{\sigma}}
\def\hbsigma{\hat{\bfsym \sigma}}

\def\hSigma{\hat{\Sigma}}
\def\hbSigma{\hat{\bfsym \Sigma}}

\def\hlambda{\hat{\lambda}}
\def\hblambda{\hat{\bfsym \lambda}}

\def\hLambda{\hat{\Lambda}}
\def\hbLambda{\hat{\bfsym \Lambda}}

\def\homega {\hat {\omega}}
\def\hbomega {\hat{\bfsym {\omega}}}

\def\hOmega {\hat {\Omega}}
\def\hbOmega {\hat{\bfsym {\Omega}}}

\def\hrho   {\hat {\rho}}
\def\hbrho {\hat{\bfsym {\rho}}}

\def\htau   {\hat {\tau}}
\def\hbtau {\hat{\bfsym {\tau}}}

\def\hxi{\hat{\xi}}
\def\hbxi{\hat{\bfsym {\xi}}}

\def\hzeta{\hat{\zeta}}
\def\hbzeta{\hat{\bfsym {\bZeta}}}

% ================
% General
% ================
\def\andd{\mbox{and}}

\def\+#1{\mathcal{#1}}
\def\-#1{\textup{#1}}

\def\br#1{\overline{#1}}

\def\set#1{\left\{ #1 \right\}}
\def\pth#1{\left( #1 \right)}
\def\bth#1{\left[ #1 \right]}
\def\abth#1{\left | #1 \right |}

\def\const{\mbox{const.}\quad}

\def\defequal {\triangleq}
\def\defeq {\coloneqq}

\newcommand{\etal}{{\it et al }}
\newcommand{\eg}{{\it e.g. }}
\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}

\def \lr {\left}
\def \rt {\right}
\def \lra {\Longrightarrow}
\def \lla {\Longleftarrow}
\def \era {\Longleftrightarrow}

\def\say{\mbox{(say)}}
\def\sgn{\mbox{sgn}}

\def\wh{\widehat}
\def\wt{\widetilde}

\newcommand{\vect}[1]{{\textup{vec}\pth{#1}}}
\newcommand{\V}[1]{{\boldsymbol{#1}}}

\def \vsp {\vspace{2mm}}
\def\diam#1{\textup{diam}\left(#1\right)}

\def \longmid {\,\middle\vert\,}


% =================
% Algebra
% =================
\newcommand{\abs}[1]{\left\lvert#1\right\rvert}

\newcommand{\diag}{\textup{diag}}
\def\dim{\textup{dim}}

\def\eigmax{\gamma_{\mathsf{max}}}
\def\eigmin{\gamma_{\textup{min}}}

\newcommand{\la}{\left \langle}
\newcommand{\ra}{\right \rangle}

\newcommand{\La}{\left\langle\kern-0.64ex\left\langle}
\newcommand{\Ra}{\right\rangle\kern-0.64ex\right\rangle}

\def\Norm#1#2{{\left\vert\kern-0.4ex\left\vert\kern-0.4ex\left\vert #1
    \right\vert\kern-0.4ex\right\vert\kern-0.4ex\right\vert}_{#2}}
\def\norm#1#2{{\left\|#1\right\|}_{#2}}

\def\lonenorm#1{\norm{#1}{1}}
\def\ltwonorm#1{\norm{#1}{2}}
%\def\fnorm#1{\Norm{#1}{\textup{F}}}
\def\fnorm#1{\norm{#1}{\textup{F}}}
\def\supnorm#1{\norm{#1}{\infty}}
\def\opnorm#1{\norm{#1}{\textup{OP}}}

\def \Null  {\textup{null}}

\def \Proj  {\mathbb{P}}

\newcommand{\rank}{\textup{rank}}
\def \range {\textup{range}}
\def \rmt   {\top}
\def \rmh   {\mathrm{H}}

\def \sigmax {\sigma_{\mathsf{max}}}
\def \sigmin {\sigma_{\textup{min}}}
\newcommand{\sign}{\textup{sign}}
\def \supp#1{\textup{supp}\left(#1\right)}

\def\tr#1{\textup{tr}\left(#1\right)}
\def \tracemgf {\br{\textup{tr}}}

%indicator
\newcommand{\1}{{\rm 1}\kern-0.25em{\rm I}}
\def\indict#1{{\rm 1}\kern-0.25em{\rm I}_{\{#1\}}}

%ceil and floor
\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}

% =================
% Analysis
% ================
\def\card#1{\left|\left\{#1\right\}\right|}
\def \Fourier {\mathcal{F}}
\def \costf   {\mathpzc{C}}
\def \dist    {\textup{dist}}
\def \interior {\textup{int}}
\def \real {\textup{Re}}
\def \imag {\textup{Im}}
\def \eps  {\epsilon}
\def \eps {\varepsilon}
\def \mps {\mapsto}
\def \diff {{\rm d}}
\def \iprod#1#2{\left\langle #1, #2 \right\rangle}
% =================
% Geometry
% =================

\def \Vol {\textup{Vol}}

\def\set#1{\left\{#1\right\}}

\def\ball#1#2#3{\bfm{B}^{#1}\left(#2; #3\right)}
\def\sphere#1#2#3{\mathbb{S}^{#1}\left(#2; #3\right)}
\def\unitsphere#1{\mathbb{S}^{#1}}
% =================
% Optimization
% ================
\newcommand{\argmax}{\textup{argmax}}
\newcommand{\argmin}{\textup{argmin}}

\def \Ind {\mathbbm{1}}
\def \St  {\textup{~s.t.~}}
\def \Epi {\textup{Epi}}
\def \dom {\textbf{dom}}
\def \prox {\mathsf{prox}}
\def \Prox {\mathsf{prox}}
\def \opt {\textup{opt}}

% =================
% Probability
% =================

\def \E {\mathbb{E}}
\def\Expect#1#2{\E_{#1}\left[#2\right]}
\def\Cov  {\textup{cov}}

\def \logdet {\log\det}
\def \normdist {\mathcal{N}}
\def \Pr {\textup{Pr}}
\newcommand{\Prob}[1]{\Pr\left[#1\right]}
\def \Var#1{\textup{Var}\left[#1\right]}
\def \as   {\textup{a.s.}}
\def \bern {\textup{Bern}}
\def \Ent  {\textup{Ent}}
\def \LogMnt#1{\log \E e^{#1}}
\def \lsim {\lesssim}
\def \gsim {\gtrsim}
\def\kl#1#2{\bb{D}\left(#1||#2\right)}

\def \iid {\stackrel{i.i.d}{\sim}}

% =================
% Statistics
% =================
\def \natu {\natural}
\def \vcap {\wedge}
\def \vcup {\vee}

% =======================
% Signal Processing
% =======================
\def \snr {\mathsf{\mathbf{snr}}}
%\def \dh  {\textup{d}_{\textup{H}}}
\def \dh {\mathsf{d_H}}
\def \entH {\mathsf{H}}
\def \enth {\mathsf{h}}
\def \entI {\textup{I}}




\newcommand{\tcr}[1]{\textcolor{red}{#1}}
%\numberwithin{equation}{section}

%equations
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\beqas}{\begin{eqnarray*}}
\newcommand{\eeqas}{\end{eqnarray*}}
\def\bal#1\eal{\begin{align}#1\end{align}}
\def\bals#1\eals{\begin{align*}#1\end{align*}}
\def\bsal#1\esal{\begin{small}\begin{align}#1\end{align}\end{small}}
\def\bsals#1\esals{\begin{small}\begin{align*}#1\end{align*}\end{small}}
\def\bsfal#1\esfal{\begin{small}\begin{flalign}#1\end{flalign}\end{small}}
\newcommand{\nn}{\nonumber}
\def\vgap{\vspace*{.1in}}

