\( % groupings of objects. \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\seq}[1]{\left(#1\right)} \newcommand{\ang}[1]{\langle#1\rangle} \newcommand{\tuple}[1]{\left(#1\right)} \newcommand{\size}[1]{\left| #1\right|} \newcommand{\comp}{\circ} % numerical shortcuts. \newcommand{\abs}[1]{\left| #1\right|} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} % linear algebra shortcuts. \newcommand{\change}{\Delta} \newcommand{\norm}[1]{\left\| #1\right\|} \newcommand{\dprod}[1]{\langle#1\rangle} \newcommand{\linspan}[1]{\langle#1\rangle} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\der}[1]{\frac{d#1}{dx}} \newcommand{\lap}{\Delta} \newcommand{\kron}{\otimes} \newcommand{\nperp}{\nvdash} \newcommand{\mat}[1]{\left[ \begin{smallmatrix}#1 \end{smallmatrix} \right]} % derivatives and limits \newcommand{\partder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\partdern}[3]{\frac{\partial^{#3 #1}}{\partial #2^{#3}}} \newcommand{\gradient}{\nabla} \newcommand{\subdifferential}{\partial} % Arrows \newcommand{\diverge}{\nearrow} \newcommand{\notto}{\nrightarrow} \newcommand{\up}{\uparrow} \newcommand{\down}{\downarrow} % gets and gives are defined! % ordering operators \newcommand{\oleq}{\preceq} \newcommand{\ogeq}{\succeq} % programming and logic operators \newcommand{\dfn}{:=} \newcommand{\assign}{:=} \newcommand{\co}{\ co\ } \newcommand{\en}{\ en\ } % logic operators \newcommand{\xor}{\oplus} \newcommand{\Land}{\bigwedge} \newcommand{\Lor}{\bigvee} \newcommand{\finish}{\Box} \newcommand{\contra}{\Rightarrow \Leftarrow} \newcommand{\iseq}{\stackrel{_?{=}}} % Set theory \newcommand{\symdiff}{\Delta} \newcommand{\setdiff}{\backslash} \newcommand{\union}{\cup} \newcommand{\inters}{\cap} \newcommand{\Union}{\bigcup} \newcommand{\Inters}{\bigcap} \newcommand{\nullSet}{\phi} % graph theory \newcommand{\nbd}{\Gamma} % Script alphabets % For reals, use \Re % greek letters \newcommand{\eps}{\epsilon} \newcommand{\del}{\delta} \newcommand{\ga}{\alpha} \newcommand{\gb}{\beta} \newcommand{\gd}{\del} \newcommand{\gp}{\pi} \newcommand{\gf}{\phi} \newcommand{\gh}{\eta} \newcommand{\gF}{\Phi} \newcommand{\gl}{\lambda} \newcommand{\gm}{\mu} \newcommand{\gn}{\nu} \newcommand{\gr}{\rho} \newcommand{\gs}{\sigma} \newcommand{\gth}{\theta} \newcommand{\gx}{\xi} \newcommand{\gw}{\omega} \newcommand{\sw}{\sigma} \newcommand{\SW}{\Sigma} \newcommand{\ew}{\lambda} \newcommand{\EW}{\Lambda} \newcommand{\Del}{\Delta} \newcommand{\gD}{\Delta} \newcommand{\gG}{\Gamma} \newcommand{\gW}{\Omega} \newcommand{\gS}{\Sigma} \newcommand{\gTh}{\Theta} % Bold english letters. \newcommand{\bA}{\mathbf{A}} \newcommand{\bB}{\mathbf{B}} \newcommand{\bC}{\mathbf{C}} \newcommand{\bD}{\mathbf{D}} \newcommand{\bE}{\mathbf{E}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bI}{\mathbf{I}} \newcommand{\bJ}{\mathbf{J}} \newcommand{\bK}{\mathbf{K}} \newcommand{\bL}{\mathbf{L}} \newcommand{\bM}{\mathbf{M}} \newcommand{\bN}{\mathbf{N}} \newcommand{\bO}{\mathbf{O}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bQ}{\mathbf{Q}} \newcommand{\bR}{\mathbf{R}} \newcommand{\bS}{\mathbf{S}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bU}{\mathbf{U}} \newcommand{\bV}{\mathbf{V}} \newcommand{\bW}{\mathbf{W}} \newcommand{\bX}{\mathbf{X}} \newcommand{\bY}{\mathbf{Y}} \newcommand{\bZ}{\mathbf{Z}} \newcommand{\bba}{\mathbf{a}} \newcommand{\bbb}{\mathbf{b}} \newcommand{\bbc}{\mathbf{c}} \newcommand{\bbd}{\mathbf{d}} \newcommand{\bbe}{\mathbf{e}} \newcommand{\bbf}{\mathbf{f}} \newcommand{\bbg}{\mathbf{g}} \newcommand{\bbh}{\mathbf{h}} \newcommand{\bbk}{\mathbf{k}} \newcommand{\bbl}{\mathbf{l}} \newcommand{\bbm}{\mathbf{m}} \newcommand{\bbn}{\mathbf{n}} \newcommand{\bbp}{\mathbf{p}} \newcommand{\bbq}{\mathbf{q}} \newcommand{\bbr}{\mathbf{r}} \newcommand{\bbs}{\mathbf{s}} \newcommand{\bbt}{\mathbf{t}} \newcommand{\bbu}{\mathbf{u}} \newcommand{\bbv}{\mathbf{v}} \newcommand{\bbw}{\mathbf{w}} \newcommand{\bbx}{\mathbf{x}} \newcommand{\bby}{\mathbf{y}} \newcommand{\bbz}{\mathbf{z}} \newcommand{\0}{\mathbf{0}} \newcommand{\1}{\mathbf{1}} % Caligraphic english alphabet \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} % Formatting shortcuts \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} \newcommand{\htext}[2]{\texorpdfstring{#1}{#2}} % Statistics \newcommand{\distr}{\sim} \newcommand{\stddev}{\sigma} \newcommand{\covmatrix}{\Sigma} \newcommand{\mean}{\mu} \newcommand{\param}{\theta} \newcommand{\gthEst}{\hat{\theta}} \newcommand{\ftr}{\phi} \newcommand{\est}[1]{\hat{#1}} % General utility \newcommand{\todo}[1]{\textbf{[TODO]}] \footnote{TODO: #1}} \newcommand{\tbc}{[\textbf{Incomplete}]} \newcommand{\chk}{[\textbf{Check}]} \newcommand{\why}{[\textbf{Find proof}]} \newcommand{\opt}[1]{\textit{#1}} \newcommand{\experience}[1]{[\textbf{Personal Experience}]: #1 \blacktriangle} \newcommand{\pf}[1]{[\textbf{Proof}]: #1 \Box} \newcommand{\core}[1]{\textbf{Core Idea}: #1 \Arrowvert} \newcommand{\example}[1]{\textbf{Example}: #1 \blacktriangle} \newcommand{\error}[1]{\textbf{Error alert}: #1 \triangle} \newcommand{\oprob}{[\textbf{OP}]: } \renewcommand{\~}{\htext{$\sim$}{~}} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator*{\argmax}{arg\,max} \)

+Optimization

Aka Mathematical programming.

Notation: Vector space V, field F.

Solver design

Make fast solvers for specific classes of problems.

Make specification frameworks (see later section).

Applications of Constrained optimization

Convex optimization problems found widely in nature.

Untrained intuition is not very good at recognizing and formulating proper optimization problems: easy problems appear hard.

+Optimization
- - +Convex prog