# Closest Farthest Widest
## 5 Discussion  
The farthest and diameter problems are natural problems of intrinsic interest. Given their non-convexity, they have not received nearly the attention in the mathematical literature as the closest problem. Exact mathematical solutions are available in some special cases. Research on fast algorithms appears to be limited to random point clouds. Infinite sets defined by mathematical formulas have been largely ignored. The current paper partially rectifies this omission and demonstrates the value of continuous optimization techniques. The Frank-Wolfe and projected gradient ascent algorithms are relatively easy to code and extremely fast in high dimensions. Our preliminary experiments tilt toward the Frank-Wolfe algorithms as the more reliable of the two options. The full Julia code for our numerical examples appears at [[https://github.com/[PERSON]/ClosestFarthestWidest](https://github.com/[PERSON]/ClosestFarthestWidest)]([https://github.com/[PERSON]/ClosestFarthestWidest](https://github.com/[PERSON]/ClosestFarthestWidest)) (accessed on 19 January 2024). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Set** & **Dimension** & **Type** & **Homotopy** & **Fraction** & **Maximum** & **Seconds** \\ \hline L1 ball & 2 & widest & yes & 0.99 & 2.0 & 0.0912 \\ elastic net & 2 & farthest & no & & 2.2883 & 0.159 \\ elastic net & 2 & widest & no & 1.0 & 1.4641 & 0.0279 \\ elastic net & 2 & widest & yes & 0.82 & 1.4641 & 0.112 \\ box & 3 & farthest & no & & 3.9572 & 0.0269 \\ box & 3 & widest & no & 1.0 & 3.4641 & 0.0246 \\ box & 3 & widest & yes & 0.99 & 3.4641 & 0.0357 \\ ball \(\cap\) orthant & 3 & farthest & no & & 3.2791 & 0.0257 \\ ball \(\cap\) orthant & 3 & widest & no & 0.9 & 1.0 & 0.0259 \\ ball \(\cap\) orthant & 3 & widest & yes & 0.77 & 1.4142 & 0.0286 \\ simplex & 3 & farthest & no & & 3.0727 & 0.0248 \\ simplex & 3 & widest & no & 0.64 & 1.4142 & 0.0258 \\ simplex & 3 & widest & yes & 0.93 & 1.4142 & 0.0291 \\ L1 ball & 3 & farthest & no & & 3.0727 & 0.0255 \\ L1 ball & 3 & widest & no & 1.0 & 2.0 & 0.025 \\ L1 ball & 3 & widest & yes & 1.0 & 2.0 & 0.0296 \\ elastic net & 3 & farthest & no & & 2.8509 & 0.0574 \\ elastic net & 3 & widest & no & 1.0 & 1.4641 & 0.0303 \\ elastic net & 3 & widest & yes & 0.74 & 1.4641 & 0.0616 \\ box & 10 & farthest & no & & 5.6758 & 0.0283 \\ box & 10 & widest & no & 1.0 & 6.3246 & 0.0314 \\ box & 10 & widest & yes & 0.95 & 6.3246 & 0.0401 \\ ball \(\cap\) orthant & 10 & farthest & no & & 3.6022 & 0.0268 \\ ball \(\cap\) orthant & 10 & widest & no & 1.0 & 1.0 & 0.0279 \\ ball \(\cap\) orthant & 10 & widest & yes & 1.0 & 1.4142 & 0.0396 \\ simplex & 10 & farthest & no & & 3.2937 & 0.0272 \\ simplex & 10 & widest & no & 0.08 & 1.0801 & 0.0271 \\ simplex & 10 & widest & yes & 0.88 & 1.4142 & 0.0327 \\ L1 ball & 10 & farthest & no & & 3.2937 & 0.0271 \\ L1 ball & 10 & widest & no & 1.0 & 2.0 & 0.0272 \\ L1 ball & 10 & widest & yes & 0.97 & 2.0 & 0.0363 \\ elastic net & 10 & farthest & no & & 3.1077 & 0.0415 \\ elastic net & 10 & widest & no & 1.0 & 1.4641 & 0.0289 \\ elastic net & 10 & widest & yes & 0.44 & 1.4641 & 0.0962 \\ box & 1000 & farthest & no & & 59.61 & 0.0515 \\ box & 1000 & widest & no & 1.0 & 63.246 & 0.0914 \\ box & 1000 & widest & yes & 0.01 & 63.019 & 1.11 \\ ball \(\cap\) orthant & 1000 & farthest & no & & 31.962 & 0.0284 \\ ball \(\cap\) orthant & 1000 & widest & no & 1.0 & 1.0 & 0.0377 \\ ball \(\cap\) orthant & 1000 & widest & yes & 1.0 & 1.4142 & 0.209 \\ simplex & 1000 & farthest & no & & 31.325 & 0.0466 \\ simplex & 1000 & widest & no & 0.02 & 1.0005 & 0.0588 \\ simplex & 1000 & widest & yes & 0.02 & 1.4142 & 0.632 \\ L1 ball & 1000 & farthest & no & & 31.325 & 0.0435 \\ L1 ball & 1000 & widest & no & 1.0 & 2.0 & 0.0722 \\ L1 ball & 1000 & widest & yes & 0.52 & 2.0 & 0.566 \\ elastic net & 1000 & farthest & no & & 31.3 & 0.609 \\ elastic net & 1000 & widest & no & 1.0 & 1.4641 & 0.29 \\ elastic net & 1000 & widest & yes & 0.03 & 1.4641 & 4.2 \\ \hline \hline \end{tabular}
\end{table}