\def\ScribeStr{??} \def\LecStr{??} \def\LecNum{??} \def\LecTitle{??} \def\LecDate{??} \newcommand{\Scribe}[1]{\def\ScribeStr{{\bf Scribe: } #1}} \newcommand{\Scribes}[1]{\def\ScribeStr{{\bf Scribes: } #1}} \newcommand{\Lecturer}[1]{\def\LecStr{{\bf Lecturer: } #1}} \newcommand{\Lecturers}[1]{\def\LecStr{{\bf Lecturers: } #1}} \newcommand{\LectureNumber}[1]{\def\LecNum{#1}} \newcommand{\LectureDate}[1]{\def\LecDate{#1}} \newcommand{\LectureTitle}[1]{\def\LecTitle{#1}} \newcommand{\VEC}{\operatorname{vec}} \newcommand{\vecc}{\operatorname{vec}} \newcommand{\svec}{\operatorname{svec}} \newcommand{\cond}{\operatorname{cond}} \newcommand{\cone}{\operatorname{cone}} \newcommand{\conv}{\operatorname{conv}} \newcommand{\argmin}{\operatorname{argmin}} \newcommand{\rk}{\operatornamewithlimits{rk}} \newcommand{\range}{\operatorname{Range}} \newcommand{\Ker}{\operatorname{Ker}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\codim}{\operatorname{co-dim}} \newcommand{\Mod}{\operatorname{Mod}} \newcommand{\Span}{\operatorname{span}} \newcommand{\jrk}[1]{\rk_{#1}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Arw}{\operatorname{Arw}} \newcommand{\Ocomm}{C_{\J}} \newcommand{\Comm}{\mathfrak{C}} \newcommand{\bigO}{\mathcal{O}} \newcommand{\Cn}{\mathcal{C}_n} \newcommand{\defeq}{\stackrel{\mathrm{def}}{=}} \newcommand{\pd}{\succ} \newcommand{\xbar}{\bar{x}} %\newcommand{\Th}[1]{$#1^\mathrm{th}$} \newcommand{\Th}[1]{#1\textsuperscript{th}} \newcommand{\ubar}{\bar{u}} \newcommand{\tlup}[1]{\widetilde{#1}} \newcommand{\tldn}[1]{\undertilde{#1}} %\newcommand{\tlup}[1]{\tilde{#1}} %\newcommand{\tldn}[1]{\undertilde{#1}} \newcommand{\bd}{\operatorname{bd}} \newcommand{\cl}{\operatorname{cl}} \newcommand{\Int}{\operatorname{Int}} \newcommand{\sgn}{\operatornamewithlimits{sgn}} \newcommand{\norminf}[1]{\left\|{#1}\right\|_\infty} \newcommand{\normone}[1]{\left\|{#1}\right\|_1} \newcommand{\normp}[1]{\left\|{#1}\right\|_p} \newcommand{\normq}[1]{\left\|{#1}\right\|_q} \newcommand{\normtwo}[1]{\left\|{#1}\right\|_2} \newcommand{\normF}[1]{\left\|{#1}\right\|_{\mathrm{F}}} \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{\tr}{\operatorname{tr}} \newcommand{\trace}{\operatorname{trace}} \newcommand{\jtr}[1]{\tr_{#1}} \newcommand{\Det}{\operatorname{Det}} \newcommand{\jdet}[1]{\det_{#1}} \newcommand{\Tr}{\operatorname{Trace}} \newcommand{\Trace}{\operatorname{Trace}} \newcommand{\op}[1]{{#1}^{\text{old}}} %old solution \newcommand{\np}[1]{{#1}^{\text{new}}} %new solution \newcommand{\Qp}[1]{\mathbf{Q}_{#1}} \newcommand{\jprod}{\circ}\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{\vh}{\mathbf{h}} \newcommand{\vi}{\mathbf{i}} \newcommand{\vj}{\mathbf{j}} \newcommand{\vk}{\mathbf{k}} \newcommand{\vl}{\mathbf{l}} \newcommand{\vm}{\mathbf{m}} \newcommand{\vn}{\mathbf{n}} \newcommand{\vo}{\mathbf{o}} \newcommand{\vp}{\mathbf{p}} \newcommand{\vq}{\mathbf{q}} \newcommand{\vr}{\mathbf{r}} \newcommand{\vs}{\mathbf{s}} \newcommand{\vt}{\mathbf{t}} \newcommand{\vu}{\mathbf{u}} \newcommand{\vv}{\mathbf{v}} \newcommand{\vw}{\mathbf{w}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vy}{\mathbf{y}} \newcommand{\vz}{\mathbf{z}} \newcommand{\valpha}{\boldsymbol{\alpha}} \newcommand{\vbeta}{\boldsymbol{\beta}} \newcommand{\vgamma}{\boldsymbol{\gamma}} \newcommand{\vdelta}{\boldsymbol{\delta}} \newcommand{\vepsilon}{\boldsymbol{\epsilon}} \newcommand{\vvarepsilon}{\boldsymbol{\varepsilon}} \newcommand{\vzeta}{\boldsymbol{\zeta}} \newcommand{\veta}{\boldsymbol{\eta}} \newcommand{\vtheta}{\boldsymbol{\theta}} \newcommand{\vvartheta}{\boldsymbol{\vartheta}} \newcommand{\viota}{\boldsymbol{\iota}} \newcommand{\vkappa}{\boldsymbol{\kappa}} \newcommand{\vlambda}{\boldsymbol{\lambda}} \newcommand{\vLambda}{\boldsymbol{\lambda}} \newcommand{\vmu}{\boldsymbol{\mu}} \newcommand{\vnu}{\boldsymbol{\nu}} \newcommand{\vxi}{\boldsymbol{\xi}} \newcommand{\vomicron}{\boldsymbol{o}} \newcommand{\vpi}{\boldsymbol{\pi}} \newcommand{\vvarpi}{\boldsymbol{\varpi}} \newcommand{\vrho}{\boldsymbol{\rho}} \newcommand{\vvarrho}{\boldsymbol{\varrho}} \newcommand{\vsigma}{\boldsymbol{\sigma}} \newcommand{\vvarsigma}{\boldsymbol{\varsigma}} \newcommand{\vtau}{\boldsymbol{\tau}} \newcommand{\vupsilon}{\boldsymbol{\upsilon}} \newcommand{\vphi}{\boldsymbol{\phi}} \newcommand{\vvarphi}{\boldsymbol{\varphi}} \newcommand{\vchi}{\boldsymbol{\chi}} \newcommand{\vpsi}{\boldsymbol{\psi}} \newcommand{\vomega}{\boldsymbol{\omega}} \newcommand{\bva}{\overline{\mathbf{a}}} \newcommand{\bvb}{\overline{\mathbf{b}}} \newcommand{\bvc}{\overline{\mathbf{c}}} \newcommand{\bvd}{\overline{\mathbf{d}}} \newcommand{\bve}{\overline{\mathbf{e}}} \newcommand{\bvf}{\overline{\mathbf{f}}} \newcommand{\bvg}{\overline{\mathbf{g}}} \newcommand{\bvh}{\overline{\mathbf{h}}} \newcommand{\bvi}{\overline{\mathbf{i}}} \newcommand{\bvj}{\overline{\mathbf{j}}} \newcommand{\bvk}{\overline{\mathbf{k}}} \newcommand{\bvl}{\overline{\mathbf{l}}} \newcommand{\bvm}{\overline{\mathbf{m}}} \newcommand{\bvn}{\overline{\mathbf{n}}} \newcommand{\bvo}{\overline{\mathbf{o}}} \newcommand{\bvp}{\overline{\mathbf{p}}} \newcommand{\bvq}{\overline{\mathbf{q}}} \newcommand{\bvr}{\overline{\mathbf{r}}} \newcommand{\bvs}{\overline{\mathbf{s}}} \newcommand{\bvt}{\overline{\mathbf{t}}} \newcommand{\bvu}{\overline{\mathbf{u}}} \newcommand{\bvv}{\overline{\mathbf{v}}} \newcommand{\bvw}{\overline{\mathbf{w}}} \newcommand{\bvx}{\overline{\mathbf{x}}} \newcommand{\bvy}{\overline{\mathbf{y}}} \newcommand{\bvz}{\overline{\mathbf{z}}} \newcommand{\bvA}{{\mathbf{A}}} \newcommand{\bvB}{{\mathbf{B}}} \newcommand{\bvC}{{\mathbf{C}}} \newcommand{\bvD}{{\mathbf{D}}} \newcommand{\bvE}{{\mathbf{E}}} \newcommand{\bvF}{{\mathbf{F}}} \newcommand{\bvG}{{\mathbf{G}}} \newcommand{\bvH}{{\mathbf{H}}} \newcommand{\bvI}{{\mathbf{I}}} \newcommand{\bvJ}{{\mathbf{J}}} \newcommand{\bvK}{{\mathbf{K}}} \newcommand{\bvL}{{\mathbf{L}}} \newcommand{\bvM}{{\mathbf{M}}} \newcommand{\bvN}{{\mathbf{N}}} \newcommand{\bvO}{{\mathbf{O}}} \newcommand{\bvP}{{\mathbf{P}}} \newcommand{\bvQ}{{\mathbf{Q}}} \newcommand{\bvR}{{\mathbf{R}}} \newcommand{\bvS}{{\mathbf{S}}} \newcommand{\bvT}{{\mathbf{T}}} \newcommand{\bvU}{{\mathbf{U}}} \newcommand{\bvV}{{\mathbf{V}}} \newcommand{\bvW}{{\mathbf{W}}} \newcommand{\bvX}{{\mathbf{X}}} \newcommand{\bvY}{{\mathbf{Y}}} \newcommand{\bvZ}{{\mathbf{Z}}} \newcommand{\hva}{{\mathbf{\hat a}}} \newcommand{\hvb}{{\mathbf{\hat b}}} \newcommand{\hvc}{{\mathbf{\hat c}}} \newcommand{\hvd}{{\mathbf{\hat d}}} \newcommand{\hve}{{\mathbf{\hat e}}} \newcommand{\hvf}{{\mathbf{\hat f}}} \newcommand{\hvg}{{\mathbf{\hat g}}} \newcommand{\hvh}{{\mathbf{\hat h}}} \newcommand{\hvi}{{\mathbf{\hat i}}} \newcommand{\hvj}{{\mathbf{\hat j}}} \newcommand{\hvk}{{\mathbf{\hat k}}} \newcommand{\hvl}{{\mathbf{\hat l}}} \newcommand{\hvm}{{\mathbf{\hat m}}} \newcommand{\hvn}{{\mathbf{\hat n}}} \newcommand{\hvo}{{\mathbf{\hat o}}} \newcommand{\hvp}{{\mathbf{\hat p}}} \newcommand{\hvq}{{\mathbf{\hat q}}} \newcommand{\hvr}{{\mathbf{\hat r}}} \newcommand{\hvs}{{\mathbf{\hat s}}} \newcommand{\hvt}{{\mathbf{\hat t}}} \newcommand{\hvu}{{\mathbf{\hat u}}} \newcommand{\hvv}{{\mathbf{\hat v}}} \newcommand{\hvw}{{\mathbf{\hat w}}} \newcommand{\hvx}{{\mathbf{\hat x}}} \newcommand{\hvy}{{\mathbf{\hat y}}} \newcommand{\hvz}{{\mathbf{\hat z}}} \newcommand{\xt}{\tlup{\vx}} \newcommand{\wt}{\tlup{\vw}} \newcommand{\st}{\tldn{\vs}} \newcommand{\ct}{\tldn{\vc}} \newcommand{\at}{\tldn{\va_i}} \newcommand{\At}{\tldn{A}} \newcommand{\Xt}{\tilde{X}} \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}{\vs^{\onehalf}} \newcommand{\sisq}{\vs^{\nonehalf}} \newcommand{\dx}{\Delta \vx} \newcommand{\dy}{\Delta \vy} \newcommand{\ds}{\Delta \vs} \newcommand{\sh}{\hat{\vs}} \newcommand{\A}{\mathcal{A}} \newcommand{\B}{\mathcal{B}} \newcommand{\Bn}{\mathcal{B}_n} \newcommand{\C}{\mathcal{C}} \newcommand{\D}{\mathcal{D}} \newcommand{\E}{\mathcal{E}} \newcommand{\F}{\mathcal{F}} \newcommand{\G}{\mathcal{G}} \newcommand{\Gp}{\mathcal{G}_+} \renewcommand{\H}{\mathcal{H}} \newcommand{\I}{\mathcal{I}} \newcommand{\J}{\mathcal{J}} \newcommand{\K}{\mathcal{K}} \newcommand{\Lo}{\mathcal{L}} \newcommand{\M}{\mathcal{M}} \newcommand{\N}{\mathcal{N}} \newcommand{\Po}{\mathcal{P}} \newcommand{\Q}{\mathcal{Q}} \newcommand{\Oc}{\mathcal{O}} \newcommand{\Sy}{\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{\matA}{\mathfrak{A}} \newcommand{\matB}{\mathfrak{B}} \newcommand{\matC}{\mathfrak{C}} \newcommand{\matD}{\mathfrak{D}} \newcommand{\matE}{\mathfrak{E}} \newcommand{\matF}{\mathfrak{F}} \newcommand{\matG}{\mathfrak{G}} \newcommand{\matH}{\mathfrak{H}} \newcommand{\matI}{\mathfrak{I}} \newcommand{\matJ}{\mathfrak{J}} \newcommand{\matK}{\mathfrak{K}} \newcommand{\matL}{\mathfrak{L}} \newcommand{\matM}{\mathfrak{M}} \newcommand{\matN}{\mathfrak{N}} \newcommand{\matO}{\mathfrak{O}} \newcommand{\matP}{\mathfrak{P}} \newcommand{\matQ}{\mathfrak{Q}} \newcommand{\matR}{\mathfrak{R}} \newcommand{\matS}{\mathfrak{S}} \newcommand{\matT}{\mathfrak{T}} \newcommand{\matU}{\mathfrak{U}} \newcommand{\matV}{\mathfrak{V}} \newcommand{\matW}{\mathfrak{W}} \newcommand{\matX}{\mathfrak{X}} \newcommand{\matY}{\mathfrak{Y}} \newcommand{\matZ}{\mathfrak{Z}} \newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} \newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} \newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} \newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} \newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} \newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} \newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} \newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} \newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} \newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} \newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} \newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} \newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} \newcommand{\real}{\mathbb{R}} \newcommand{\realS}{\mathbb{S}} \newcommand{\realH}{\mathbb{H}} \newcommand{\realQ}{\mathbb{Q}} \newcommand{\realO}{\mathbb{O}} \newcommand{\ST}{{\text{s.t.}}} \newcommand{\NF}[1]{\N_{F/{#1}}} \newcommand{\Quart}{\mbox{\boldmath H}} \newcommand{\psd}{\succcurlyeq} \newcommand{\pdef}{\succ} %\newcommand{\pd}{\succ} %use for strict semidefinite inequality \newcommand{\Qgeq}{\succcurlyeq_\Q} \newcommand{\Qgt}{\succ_\Q} \newcommand{\Kgeq}{\succcurlyeq_\K} \newcommand{\Kgt}{\succ_\K} \newcommand{\skron}{\circledast} \newcommand{\kron}{\otimes} \newcommand{\Diag}{\operatorname{Diag}} \newcommand{\diag}{\mathbf{diag}} \newcommand{\dt}{\Delta} \DeclareMathOperator*{\Bd}{bd} %For boundary \DeclareMathOperator*{\Cl}{cl}%For closure %Use the following for definitions lemmas etc. \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma} \newtheorem{note}{Note} \newtheorem{proposition}{Proposition} \newtheorem{theorem}{Theorem} \newtheorem{claim}{Claim} \newtheorem{remark}{Remark} \newtheorem{example}{Example} \newtheorem{corollary}{Corollary} \newtheorem{observation}{Observation} \newtheorem{fact}{Fact} \newenvironment{Example}[1]{\begin{example}[#1]\rm}{{\hfill\rule{2mm}{2mm}\par}\end{example}} \newenvironment{Remark}{\begin{remark}\rm}{{\hfill\rule{2mm}{2mm}\par}\end{remark}} \newenvironment{proof}{{\noindent \textbf{Proof:}}}{\hfill\rule{2mm}{2mm}\par} %