% The map: optimization principles placed in the hierarchy of subsystems of second-order arithmetic.
\begin{figure}[t]
\centering
\begin{tikzpicture}[
  font=\scriptsize,
  sys/.style={font=\small, anchor=east, inner sep=2pt},
  item/.style={draw=#1!55!black, fill=#1!7, rounded corners=2pt, inner sep=2pt, align=center,
               text width=4.15cm, minimum height=0.74cm,
               execute at begin node={\hyphenpenalty=10000\relax}},
  item/.default=gray,
  new/.style={item=orange, line width=0.8pt},
  bound/.style={item, dashed},
  head/.style={font=\footnotesize\itshape, text=black!70, anchor=south},
  band/.style={fill=black!3, draw=none},
]
\def\yR{0}\def\yW{1.9}\def\yA{3.75}\def\yT{5.6}\def\yP{7.0}
\def\xE{2.75}\def\xC{7.15}\def\xD{11.55}
\fill[band] (0.45,\yR-0.95) rectangle (13.75,\yR+0.95);
\fill[band] (0.45,\yA-0.95) rectangle (13.75,\yA+0.95);
\fill[band] (0.45,\yP-0.5) rectangle (13.75,\yP+0.5);
\node[sys] at (0.4,\yR) {$\RCA$};
\node[sys] at (0.4,\yW) {$\WKL$};
\node[sys] at (0.4,\yA) {$\ACA$};
\node[sys] at (0.4,\yT) {$\ATR$};
\node[sys] at (0.4,\yP) {$\PiCA$};
\node[head] at (\xE,7.55) {existence};
\node[head] at (\xC,7.55) {convergence \& selection};
\node[head] at (\xD,7.55) {duality \& combinatorics};
% RCA0
\node[new]  at (\xE,\yR+0.45) {convex minimization on $[0,1]$;\\ on $[0,1]^n$ given a modulus};
\node[item] at (\xE,\yR-0.45) {IVT; polynomial optimization\\ of fixed format (folklore)};
\node[item] at (\xC,\yR+0.45) {rates for $f(x_k)$ and $\nabla f(x_k)$};
\node[item] at (\xC,\yR-0.45) {rates from moduli of\\ uniqueness or regularity};
\node[new]  at (\xD,\yR+0.45) {LP duality and minimax,\\ all sizes};
\node[item] at (\xD,\yR-0.45) {matroid bases\\ (greedy algorithm)};
% WKL0
\node[item] at (\xE,\yW) {extreme value theorem;\\ Brouwer; Peano};
\node[item] at (\xD,\yW) {Hahn--Banach; separation of\\ open convex sets; Dilworth};
% ACA0
\node[item] at (\xE,\yA+0.45) {Bolzano--Weierstrass; lsc\\ minima; Hilbert projections};
\node[item] at (\xE,\yA-0.45) {existence of moduli\\ of regularity};
\node[item] at (\xC,\yA+0.45) {gradient descent, PPA, KM\\ converge (on $\R$)};
\node[bound] at (\xC,\yA-0.45) {Halpern, Tikhonov: projection\\ selection (implies $\ACA$)};
\node[item] at (\xD,\yA+0.45) {strict separation from a\\ separably closed set ($\R^2$)};
\node[item] at (\xD,\yA-0.45) {Hall (locally finite); shortest\\ paths; minimum spanning trees};
% ATR0
\node[item]  at (\xD,\yT+0.4) {K\H{o}nig duality\\ (countable bipartite graphs)};
\node[bound] at (\xD,\yT-0.4) {Menger: at least $\ATR$,\\ exact strength open};
\node[item] at (\xE,\yT) {Caristi for Borel maps\\ (strictly between $\ATR$ and $\PiCA$)};
% Pi11
\node[item] at (\xE,\yP) {Ekeland's variational principle};
\node[item] at (\xD,\yP) {extended K\H{o}nig and\\ Menger theorems};
% arrows
\draw[-{Latex[length=1.8mm]}, thick, orange!70!black] (\xE,\yW-0.4) -- (\xE,\yR+0.85)
   node[midway, right, font=\scriptsize\itshape, text=orange!50!black] {convexity, tameness};
\draw[-{Latex[length=1.8mm]}, thick, blue!60!black] (\xC,\yR+0.85) -- (\xC,\yA-0.85)
   node[midway, right, font=\scriptsize\itshape, text=blue!50!black, align=left] {from values\\ to iterates};
\end{tikzpicture}
\caption{The map. Principles of optimization placed at the subsystem of second-order arithmetic to which they are
equivalent over $\RCA$ (in the bottom row: provable in $\RCA$). Solid boxes are equivalences or provability results,
dashed boxes are one-sided bounds, and orange boxes are results proved in this paper (\cref{thm:convex-rca,thm:lp-rca}).
The Weihrauch counterparts of the rows are, roughly: non-uniformly computable problems such as $\IVT$ and $\XC_n$
($\RCA$), closed choice $\Cch_{[0,1]}\seqv\WKLp$ ($\WKL$), and $\lm$ ($\ACA$). Several placements depend on how sets
and functions are coded; see the text for exact statements and sources.}
\label{fig:map}
\end{figure}