# Closest Farthest Widest
## 4 Numerical Experiments  
We tested the Frank-Wolfe and projected gradient ascent algorithms on five compact convex sets: (a) the box \([-\mathbf{1},\mathbf{1}]=\{\mathbf{x}:\|\mathbf{x}\|_{\infty}\leq 1\}\), (b) the intersection of the unit ball and the non-negative orthant, (c) the probability simplex, (d) the \(\ell_{1}\) unit ball, and (e) the sublevel set \(\{\mathbf{x}:\|\mathbf{x}\|_{1}+\frac{1}{2}\|\mathbf{x}\|^{2}\leq 1\}\) determined by the elastic net penalty. All five sets are permutationally invariant. Sets (a), (d), and (e) are even. These examples are representative, and all five exact diameters are available for comparison. Tables 1 and 2 present our findings. The column headed \"Fraction\" records the fraction of the 100 trials that achieve the maximum objective. The column headed \"Seconds\" records cumulative execution times over 100 trials. Homotopy offers no advantage in the farthest problem and is limited to the diameter problem. The farthest problem is initialized by \(-\mathbf{p}\) under Frank-Wolfe and by \(P_{\mathbb{C}}(-\mathbf{p})\) under projected gradient ascent. Thus, all 100 trials are identical in each scenario. The diameter problem is initialized by two random opposing points on the unit sphere. Given the nature of the support and projection maps, all subsequent iterations fall within the set \(\mathbb{C}\). Both the Frank-Wolfe algorithms and the projected gradient descent algorithms perform well. They attain the same solutions, which in the diameter problems are identical to the known solution. Contrary to our expectation, projected gradient ascent is not noticeably slower than Frank-Wolfe. Homotopy over 11 intervening points rescues projected gradient descent on the diameter problem for sets (b) and (c). Homotopy is noticeably slower than the unadorned algorithms. The elastic net problem tends to take the most time owing to the inefficiency of bisection. Computation times scale well when the dimension \(d\) of the ambient space reaches 1000. All computations were carried out on a MacBook Pro with a 2.3 GHz 8-core i9 chip and 16 GB of memory. Although the algorithms are embarrassingly parallel across trials, our Julia code is completely serial. On the basis of reliability, these problems favor Frank-Wolfe. Although Frank-Wolfe algorithms possess a theoretically faster rate of convergence than projected gradient ascent, in the empirical trials, the two methods are quite comparable in speed. The \"Seconds\" columns of Tables 1 and 2 have respective means of 0.260 and 0.240 and respective medians of 0.065 and 0.038. The corresponding standard deviations 1.192 and 0.697 are relatively high, so for this and other obvious reasons, one should probably not read too much into these crude comparisons. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Set** & **Dimension** & **Type** & **Homotopy** & **Fraction** & **Maximum** & **Seconds** \\ \hline box & 2 & farthest & no & & 2.9624 & 0.13 \\ box & 2 & widest & no & 1.0 & 2.8284 & 0.0708 \\ box & 2 & widest & yes & 1.0 & 2.8284 & 0.0709 \\ ball \(\cap\) orthant & 2 & farthest & no & & 2.5994 & 0.0817 \\ ball \(\cap\) orthant & 2 & widest & no & 0.52 & 1.4142 & 0.0756 \\ ball \(\cap\) orthant & 2 & widest & yes & 0.52 & 1.4142 & 0.0612 \\ simplex & 2 & farthest & no & & 2.5465 & 0.0734 \\ simplex & 2 & widest & no & 1.0 & 1.4142 & 0.0697 \\ simplex & 2 & widest & yes & 1.0 & 1.4142 & 0.0707 \\ L1 ball & 2 & farthest & no & & 2.5465 & 0.0754 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Evaluation of the Frank–Wolfe algorithms.