Closest Farthest Widest

Abstract

The current paper proposes and tests algorithms for finding the diameter of a compact convex set and the farthest point in the set to another point. For these two nonconvex problems, I construct Frank–Wolfe and projected gradient ascent algorithms. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, I investigate a homotopy method that gradually deforms a ball into the target set. Motivated by the Frank–Wolfe algorithm, I also find the support function of the intersection of a convex cone and a ball centered at the origin and elaborate a known bisection algorithm for calculating the support function of a convex sublevel set. The Frank–Wolfe and projected gradient algorithms are tested on five compact convex sets: (a) the box whose coordinates range between −1 and 1, (b) the intersection of the unit ball and the non-negative orthant, (c) the probability simplex, (d) the Manhattan-norm unit ball, and (e) a sublevel set of the elastic net penalty. Frank–Wolfe and projected gradient ascent are about equally fast on these test problems. Ignoring homotopy, the Frank–Wolfe algorithm is more reliable. However, homotopy allows projected gradient ascent to recover from its failures.

Keywords

• convex set
• diameter
• farthest point
• Frank-Wolfe
• homotopy
• projected gradient ascent